Release v1.3.0: audited AMR research metadata

README.md

@@ -45,7 +45,7 @@ Paper: "[Open Mathematical Problems as an AI Reasoning Benchmark](https://huggin
 
 - **Total Problems**: 5426
 - **Erdős Problems**: 632 problems with citations and references
-- **Version**: 1.2.0
+- **Version**: 1.3.0
 - **Categories**: 17 mathematical domains
 - **Difficulty Levels**: 5 (L1: Tractable → L5: Millennium Prize)
 - **Problem Sets**: 13 curated collections
@@ -60,6 +60,17 @@ This release adds problems collected from public source lists in the AMR index.
 
 Every AMR record retains its source URL, extraction method, and status/rights review notes in the `background` field.
 
+### New in v1.3.0
+
+This release adds a **research status audit** for every AMR open-problem record. Each of the 3,342 AMR problems was investigated by an AI research fleet (literature triage with web-verified citations, solution attempts, and partial progress), and every report was then re-checked in a supervised verification pass (statement alignment, citation-fabrication screening, classification normalization, difficulty assignment). Each AMR problem now carries: a `research_classification` (`SOLVED-BY-YOU`, `SOLVED-IN-LITERATURE`, `PARTIAL-PROGRESS`, `OPEN-TRIAGE`), an updated `status` derived from the classification, an optional `research_difficulty_suggested`, and a `research_summary`. AMR difficulty levels are now differentiated across L2–L5 (previously all L3). The structured per-problem research notes (problem, literature status, work done, result, what remains) are provided as a new `research_results.json` file, and the full individual reports live in the `research/` directory (`research/<problem_number>.md`).
+
+- **Research audit**: 3,342 AMR problems classified; 183 solved (181 in the literature, 2 by the AI fleet), 963 partial progress, 2,196 open after triage.
+- **New file**: `research_results.json` — structured per-problem research notes.
+- **New directory**: `research/` — full per-problem research reports in Markdown.
+- **New fields** per AMR problem: `research_classification`, `research_summary`, `research_difficulty_suggested`.
+
+*Caveat:* classifications and summaries are machine-generated research aids, not peer-reviewed results; `SOLVED-*` entries were citation-checked, but independent verification is recommended before citing.
+
 ### Supported Tasks
 
 - Mathematical research and exploration
@@ -80,6 +91,8 @@ The dataset consists of multiple JSON files:
 4. **sets.json** - Problem set metadata (Millennium Prize, Hilbert's 23, etc.)
 5. **dataset.json** - Combined file with all data
 6. **statistics.json** - Dataset statistics
+7. **research_results.json** - Per-problem AMR research notes (status, literature, result, what remains)
+8. **research/** - Full per-problem AMR research reports as Markdown files (`research/<problem_number>.md`)
 
 ### Data Fields
 
@@ -104,6 +117,9 @@ Each problem contains:
 - `solved_by` (string, optional): Solver's name
 - `prize_amount` (int, optional): Prize money (USD)
 - `created_at` (string): Timestamp
+  - `research_classification` (string, optional): AMR research verdict, e.g. `SOLVED-IN-LITERATURE`, `PARTIAL-PROGRESS`, `OPEN-TRIAGE`
+  - `research_summary` (string, optional): 2–3 paragraph summary of the research findings
+  - `research_difficulty_suggested` (string, optional): suggested difficulty level when it differs from the default
 
 #### Categories
 
@@ -138,11 +154,11 @@ Each problem contains:
 
 ### Problems by Difficulty
 
-- L3: Advanced: 3838
+- L3: Advanced: 3440
 - L1: Tractable: 916
-- L2: Intermediate: 316
-- L4: Expert: 220
-- L5: Millennium Prize: 136
+- L2: Intermediate: 332
+- L4: Expert: 601
+- L5: Millennium Prize: 137
 
 ### Problems by Category
 
@@ -166,9 +182,9 @@ Each problem contains:
 
 ### Problems by Status
 
-- Open: 5156
-- Solved: 9
-- Partially Solved: 261
+- Open: 4271
+- Solved: 192
+- Partially Solved: 963
 
 ## Usage
 
@@ -281,5 +297,5 @@ For questions, issues, or contributions:
 
 ---
 
-**Generated**: 2026-07-31T18:36:34.574Z
-**Version**: 1.2.0
+**Generated**: 2026-08-06T00:00:00Z
+**Version**: 1.3.0

dataset.json

@@ -1,3 +1,3 @@
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+size 14990733

problems.json

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+size 15302727

research/AMR-005-0001.md

@@ -0,0 +1,46 @@
+---
+id: AMR-005-0001
+classification: SOLVED-IN-LITERATURE
+wording_corrected: yes
+---
+
+# AMR-005-0001 — Commuting billiard ball maps
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Problem 1 in the list). The original wording, verified against the published article:
+
+> Consider two nested convex domains. Then one has two billiard ball maps, $T_1$ and $T_2$, acting on the oriented lines that intersect both domains. If the domains are bounded by confocal ellipses, then the respective billiard ball maps commute. Assume that the two maps commute: $T_1 \circ T_2 = T_2 \circ T_1$.
+> **Conjecture.** The two domains are bounded by confocal ellipses.
+> For outer (a.k.a. dual) billiards, an analogous fact is proved in Tabachnikov (1994). For piece-wise analytic billiards, this conjecture was proved by Glutsyuk (2014). Of course, this problem has a multi-dimensional version, open both for inner and outer billiards.
+
+Correction made to the garbled transcription: the list version merged the 2-dimensional conjecture with the multidimensional question into a single imperative sentence and omitted the status remarks (planar dual-billiard case already solved in 1994; piecewise-analytic case solved by Glutsyuk in 2014). The transcription's mathematical content is otherwise faithful.
+
+## Status / Literature
+
+All references below were verified via Crossref, the arXiv API, and publisher pages (abstracts seen verbatim).
+
+- **Planar inner billiards — SOLVED.** A. Glutsyuk, "On 4-reflective complex analytic planar billiards", *J. Geom. Anal.* 27 (2017), 183–238 (online 2016), DOI 10.1007/s12220-016-9679-x, arXiv:1405.5990. The published abstract states that the paper provides "solutions of Tabachnikov's Commuting Billiard Conjecture ... in two dimensions; the boundary is required to be piecewise $C^4$-smooth."
+- **Higher-dimensional inner billiards — SOLVED.** A. Glutsyuk, "On commuting billiards in higher-dimensional spaces of constant curvature", *Pacific J. Math.* 305 (2020), 577–595, DOI 10.2140/pjm.2020.305.577, arXiv:1807.10567. Abstract (seen verbatim): "We consider two nested billiards in $\mathbb{R}^d$, $d\ge 3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov." The higher-dimensional case is deduced from Marcel Berger's classical theorem that in dimension $\ge 3$ only quadrics may have caustics; the paper also proves versions of Berger's theorem and the commuting result in space forms (constant curvature).
+- **Planar outer (dual) billiards — SOLVED already in 1994.** S. Tabachnikov, "Commuting dual billiard maps", *Geom. Dedicata* 53 (1994), 57–68, DOI 10.1007/BF01264044. Abstract (seen verbatim): "...We prove that if two curves are given, such that the corresponding dual billiard transformations commute, then the curves are concentric homothetic ellipses." (Note the dual-billiard answer is *concentric homothetic* ellipses, not confocal — dual billiard maps are affinely covariant.)
+- **Higher-dimensional outer billiards — apparently OPEN.** Multidimensional dual billiards exist (symplectic setting in $\mathbb{R}^{2n}$, Tabachnikov, "On the dual billiard problem", *Adv. Math.* 115 (1995), 221–249), but I found no published resolution of the commuting question there; a Crossref/arXiv search (2015–present) for commuting higher-dimensional dual/outer billiard maps returned nothing relevant.
+
+## Work done
+
+- Retrieved the original statement from the published AMJ article (link.springer.com/article/10.1007/s40598-014-0001-3) and corrected the garbled list wording.
+- Verified every citation above against Crossref metadata and, where possible, publisher abstracts (Springer page for Tabachnikov 1994; arXiv abstracts for Glutsyuk 1405.5990 and 1807.10567, including journal references).
+- Searched for post-2015 work on the multidimensional outer-billiard commuting question via the arXiv API ("outer billiard" AND commuting: 0 hits) and Crossref (no relevant result).
+
+## Result
+
+The problem is solved in the literature, with one sub-case apparently still open:
+
+1. **Planar inner case:** commuting billiard ball maps of two nested convex domains with piecewise $C^4$-smooth boundaries $\Rightarrow$ confocal ellipses (Glutsyuk 2017). The proof goes through complexified billiards: commuting forces a 4-reflective complex analytic pseudo-billiard structure near the curves, and the classification of 4-reflective germs forces the curves to be confocal conics.
+2. **Higher-dimensional inner case ($d\ge3$):** commuting actions by reflections for nested strictly convex $C^2$ billiards $\Rightarrow$ confocal ellipsoids (Glutsyuk 2020), via Berger's theorem (in dimension $\ge 3$ only quadrics admit caustics); also extended to spaces of constant curvature.
+3. **Planar outer case:** commuting dual billiard maps $\Rightarrow$ concentric homothetic ellipses (Tabachnikov 1994) — predates the list.
+4. **Higher-dimensional outer case:** no resolution found; appears to remain open.
+
+## What remains
+
+- The multidimensional commuting question for **outer/dual billiards** (symplectic dual billiard maps in $\mathbb{R}^{2n}$) seems unresolved; nothing in the literature post-2015 addresses it as far as I could verify. A natural conjecture would be: commuting dual billiard maps of nested strictly convex hypersurfaces $\Rightarrow$ concentric homothetic ellipsoids.
+- In the planar inner case the published solution assumes piecewise $C^4$ regularity; whether $C^2$ (or lower) smoothness suffices in dimension 2 is a residual regularity question (dimension $\ge 3$ needs only $C^2$, thanks to Berger-type rigidity).

research/AMR-005-0002.md

@@ -0,0 +1,69 @@
+---
+id: AMR-005-0002
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-005-0002 — Coexistence of one-parameter families of p- and q-periodic billiard trajectories
+
+## Problem (corrected statement if needed)
+
+Source: Serge Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1) (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (verified via Crossref), §2, Problem 1. The dataset transcription matches the published wording essentially verbatim:
+
+> **Problem 1.** Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories (for $p\neq q$)?
+
+Context given in the source: a curve of constant width admits a one-parameter family of 2-periodic (back-and-forth) trajectories; for every $p\ge 3$ there exist non-elliptic billiard tables admitting a one-parameter family of $p$-periodic trajectories (Baryshnikov–Zharnitsky, Math. Res. Lett. 13 (2006), 587–598, DOI 10.4310/MRL.2006.v13.n4.a8, verified via the Crossref reference list of the source article). The simplest case: does any curve of constant width, other than the circle, admit a one-parameter family of 3-periodic trajectories? The source then adds: "A similar question can be asked about outer billiards." No correction to the transcription is needed.
+
+Two clarifying remarks (standard, and consistent with the source's intent):
+
+- A one-parameter family of $p$-periodic trajectories is an invariant circle $\Gamma$ of the billiard map $T$ in the phase cylinder, with rotation number $k/p$ ($\gcd(k,p)=1$), on which $T^p=\mathrm{id}$. For $p\ge 3$ this is an *integrable rational caustic*; for $p=2$ it is a circle of fixed points of $T$.
+- In an ellipse, the 2-periodic orbits (the two axes) are isolated — they do **not** form a family. So ellipses have families for every $p\ge 3$ (Poncelet porism) but not for $p=2$; the circle additionally has a 2-periodic family (it has constant width). The problem is therefore interesting already for $(p,q)=(2,3)$, where the conjectured answer "no non-circular constant-width curve has a 3-periodic family" characterizes the circle, not arbitrary ellipses.
+
+## Status / Literature
+
+The problem in full generality is **open**. It is a weakening of the Birkhoff–Poritsky conjecture (integrable convex billiards are ellipses), itself still open in general. Verified relevant literature:
+
+- **Bialy, M., "Convex billiards and a theorem by E. Hopf", Math. Z. 214(1) (1993), 147–154, DOI 10.1007/BF02572397** (bibliographic data verified from the publisher-asserted reference lists of two Crossref-verified papers below). If the whole phase cylinder is foliated by non-contractible invariant circles, the table is a disk. This settles the extreme case "families of all periods" but not two isolated periods.
+- **Avila, A., De Simoi, J., Kaloshin, V., "An integrable deformation of an ellipse of small eccentricity is an ellipse", Ann. of Math. 184(2) (2016), 527–558, DOI 10.4007/annals.2016.184.2.5** (verified via the Crossref-verified reference list of Glutsyuk–Shustin below). Infinitesimal/one-parameter-deformation rigidity of ellipses of small eccentricity under preservation of caustics near the boundary.
+- **Kaloshin, V., Sorrentino, A., "On the local Birkhoff conjecture for convex billiards", Ann. of Math. 188(1) (2018), 315–380, DOI 10.4007/annals.2018.188.1.6** (verified via Crossref). Any $C^\infty$ billiard sufficiently close to a given ellipse that admits an integrable rational caustic of rotation number $1/q$, $q\ge 3$ (i.e., a one-parameter family of $q$-periodic orbits), is an ellipse. Hence *locally near ellipses even one family of period $\ge 3$ already forces ellipticity* — a much stronger local statement than the two-periods question.
+- **Kaloshin, V., Koudjinan, C. E., "Non co-preservation of the $1/2$ & $1/(2l+1)$-rational caustics along deformations of circles", arXiv:2107.03499** (verified via the arXiv API). Every deformation of a circle preserving both the $1/2$- and the $1/(2l+1)$-rational caustics is trivial (similarities only). This is exactly the deformational (infinitesimal) version of the $(2,\,\text{odd})$ case of Problem 1, including the "simplest case" $(2,3)$ highlighted by Tabachnikov.
+- **Bialy, M., Mironov, A. E., "The Birkhoff–Poritsky conjecture for centrally-symmetric billiard tables", Ann. of Math. 196(1) (2022), 389–413, DOI 10.4007/annals.2022.196.1.2** (verified via Crossref; note the article number is `.1.2`). For $C^2$ centrally symmetric tables, a 1/4-rotation-number invariant circle (a family of 4-periodic orbits) together with a $C^0$-foliation of the region between it and the boundary by invariant curves forces an ellipse. Global (not local), but the hypothesis is stronger than two isolated periodic families.
+- **Koval, I., "Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse", arXiv:2111.12171** (verified via the arXiv API). Local rigidity of almost every ellipse under the stronger hypothesis of integrability near the boundary (rational caustics of all rotation numbers $p/q\le 1/q_0$).
+
+Outer billiards (the analogous question is also open in full generality):
+
+- **Tabachnikov, S., "On algebraically integrable outer billiards", Pacific J. Math. 235(1) (2008), 101–104, DOI 10.2140/pjm.2008.235.89** (verified via Crossref reference lists). If the outer billiard map admits a non-constant algebraic first integral (in a real-analytic sense near the curve), the curve is an ellipse.
+- **Glutsyuk, A., Shustin, E., "On polynomially integrable planar outer billiards and curves with symmetry property", Math. Ann. 372(3–4) (2018), 1481–1501, DOI 10.1007/s00208-018-1726-4** (verified via Crossref). Every polynomially integrable planar outer billiard is elliptic — the solution of the polynomial/algebraic version of the outer-billiard integrability problem.
+- **Bialy, M., "Integrable outer billiards and rigidity", arXiv:2306.12494** (verified via the arXiv API; journal version announced 2024). If the vicinity of a smooth convex plane curve $\gamma$ of positive curvature is foliated by continuous curves invariant under the outer billiard map, then $\gamma$ is an ellipse (outer-billiard analogue of Bialy's 1993 Hopf-type rigidity, via a new generating function and the Blaschke–Santaló inequality).
+
+I found **no published result that settles either the inner or the outer two-periods question as stated**; all known results either are local (near an ellipse/circle), deformational, or assume a full foliation / full integrability, which is strictly stronger than two isolated periodic families.
+
+## Work done
+
+- Located and read the source article (AMJ open HTML) and confirmed the dataset wording matches published Problem 1 (§2) — `wording_corrected: no`.
+- Ran targeted web searches for (i) direct attacks on the two-periods problem, (ii) the constant-width/3-periodic sub-case, (iii) outer-billiard analogues.
+- Verified every citation above against Crossref (`api.crossref.org/works/<DOI>`) or the arXiv API; the Avila–De Simoi–Kaloshin, Bialy 1993, and Tabachnikov 2008 entries were cross-verified through publisher-asserted reference lists inside Crossref-verified records. One initially guessed DOI for Bialy–Mironov (`.196.1.5`) returned 404 and was corrected to `.196.1.2` — only the verified DOI is cited.
+- Mathematical analysis (no computation used):
+
+*Reformulation and reduction of the $p=2$ case.* A one-parameter family of 2-periodic orbits is a circle of fixed points of the billiard map projecting onto the whole boundary. Through every boundary point there is then a chord orthogonal to the boundary at both endpoints, and the involution swapping its endpoints is the antipodal map; equality of the two support-line distances along every direction forces the curve to have **constant width**. Conversely every constant-width curve has such a family (all diameters are double normals). This classical reduction is exactly the premise stated by Tabachnikov.
+
+*Constant perimeter lemma.* For **any** smooth one-parameter family $x(t)=(x_0(t),\dots,x_{p-1}(t))$ of $p$-periodic billiard trajectories, the perimeter $L(t)=\sum_i |x_{i+1}(t)-x_i(t)|$ is constant. Proof: writing $h(x,y)=|x-y|$ for the generating function, the billiard reflection law gives $\partial_2 h(x_{i-1},x_i)=-s_i$ and $\partial_1 h(x_i,x_{i+1})=s_i$ (the outgoing/incoming "momenta"), so $dL=\sum_i(\partial_1 h(x_i,x_{i+1})+\partial_2 h(x_{i-1},x_i))\,dx_i=\sum_i (s_i-s_i)\,dx_i=0$. (This is the classical reason Poncelet families have constant perimeter.) Hence the data of Problem 1 include two marked constants $L_p, L_q$ (plus the width $w$ when $p=2$, with $L_2=2w$), and the associated invariant circles are Lagrangian circles of rational rotation number in the phase cylinder.
+
+*Dynamical consequence.* Between the two invariant circles $\Gamma_{1/p}$, $\Gamma_{1/q}$ the billiard map is a Birkhoff twist map; Aubry–Mather theory yields Birkhoff periodic orbits of every intermediate rotation number and Mather sets for irrational ones. So the two-family hypothesis generates rich structure "in between" — but no contradiction, and the circles need not belong to a foliation: the gap to the Bialy/Bialy–Mironov-type hypotheses is exactly the missing foliation.
+
+## Result
+
+The problem is **not solved**, and I could not solve it; the honest classification is partial progress via reformulation plus a precise map of how close the literature comes:
+
+1. **Near ellipses the answer is "no" in a strong sense** (Kaloshin–Sorrentino 2018): a single one-parameter family of $q$-periodic orbits, $q\ge 3$, already characterizes ellipses locally among $C^\infty$ tables. So any counterexample to Problem 1 must be far (in a $C^\infty$ sense) from every ellipse.
+2. **The simplest case $(2,3)$ is deformationally settled near the circle** (Kaloshin–Koudjinan 2021): no non-trivial deformation of the circle preserves both the 2-periodic family (constant width to first order) and the 3-periodic family; the same holds for $(2, 2l+1)$. Thus a non-circular constant-width curve with a 3-periodic family, if it exists, is isolated from the circle in a deformation sense.
+3. **Global results all need strictly stronger hypotheses**: full foliation of the phase cylinder (Bialy 1993 $\Rightarrow$ disk), or a 1/4-caustic plus foliation below it with central symmetry (Bialy–Mironov 2022 $\Rightarrow$ ellipse). Two isolated rational invariant circles are not known to force a foliation — this is precisely the open gap.
+4. **Outer billiards**: the algebraic/polynomial integrability versions are solved (Tabachnikov 2008; Glutsyuk–Shustin 2018: only ellipses), and full integrability near the curve is solved (Bialy 2023: only ellipses); the exact two-periods question remains open there as well.
+5. Elementary but useful contributions recorded above: the $p=2$ $\Leftrightarrow$ constant-width reduction, and the constant-perimeter lemma for any one-parameter family of periodic orbits, which packages the hypothesis into two rotation numbers and two marked action constants.
+
+## What remains
+
+- The $(2,3)$ case globally: does a (smooth, strictly convex) constant-width curve other than the circle admit a one-parameter family of 3-periodic trajectories? Open. Natural approaches: (a) extend the Bialy–Mironov integral-geometry/Hopf-rigidity machinery from the 1/4-caustic to the pair (1/2-, 1/3-caustics), exploiting that constant width gives an explicit circle of fixed points; (b) Fourier/collision-operator analysis near constant-width curves generalizing the Kaloshin–Koudjinan deformation computation beyond the circle.
+- The general $(p,q)$ case for $p,q\ge 3$ globally, without proximity to an ellipse: open. Key obstacle: two rational invariant circles do not imply a foliation of the annulus between them; Birkhoff zones of instability may a priori occur there.
+- Outer-billiard two-periods question: open; even a deformational analogue of Kaloshin–Koudjinan for outer billiards seems to be missing, and Bialy's new generating function for outer billiards (arXiv:2306.12494) is a plausible tool.
+- A related open direction suggested by the literature: whether the two-period hypothesis implies rational integrability near the boundary (then Koval's local strong Birkhoff result would apply, settling the problem near almost every ellipse under any finite number of periods).

research/AMR-005-0003.md

@@ -0,0 +1,57 @@
+---
+id: AMR-005-0003
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-005-0003 — Birkhoff's theorem for Lorentz billiards
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Section 3, "Birkhoff's Theorem for Lorentz Billiards"). Original wording, verified verbatim against the published article (publisher HTML and the Springer-final PDF):
+
+> The classical Birkhoff theorem states that, for every $n\ge 3$ and $1\le k\le n/2$, the billiard system inside a plane oval has at least two $n$-periodic trajectories with the rotation number $k$. Consider the billiard system inside an oval in the Lorentz plane with the pseudo-Euclidean metric $ds^2=dx^2-dy^2$. Is there an analog of Birkhoff's theorem in this set-up?
+> Billiard trajectories in pseudo-Euclidean space can be of three types: space-like, time-like, and light-like, see Khesin and Tabachnikov (2009) for Lorentz billiards. One would expect separate existence statements for space-like and time-like trajectories.
+> A convex body in $\mathbb{R}^n$ has at least $n$ diameters (2-periodic billiard trajectories). If the ambient space is pseudo-Euclidean, $\mathbb{R}^{p,q}$, then there are at least $p$ space- and at least $q$ time-like diameters (Khesin and Tabachnikov 2009). A lower bound on the number of periodic billiard trajectories in multi-dimensional Euclidean space is obtained in Farber and Tabachnikov (2002). What happens with multi-dimensional pseudo-Euclidean billiards?
+
+The dataset transcription is a faithful condensation of this; no correction needed.
+
+## Status / Literature
+
+All references verified against Crossref metadata (DOIs below) or the arXiv API.
+
+- **Foundation: pseudo-Riemannian billiards, and the $n=2$ case.** B. Khesin, S. Tabachnikov, "Pseudo-Riemannian geodesics and billiards", *Adv. Math.* 221 (2009), 1364–1396, DOI 10.1016/j.aim.2009.02.010. Develops the symplectic/variational formalism for billiards in pseudo-Euclidean spaces; proves that a convex body in $\mathbb{R}^{p,q}$ has at least $p$ space-like and at least $q$ time-like diameters (2-periodic orbits). In the Lorentz plane this gives one space-like and one time-like 2-periodic trajectory — the first case of the desired Birkhoff analog.
+- **Euclidean multidimensional benchmark.** M. Farber, S. Tabachnikov, "Topology of cyclic configuration spaces and periodic orbits of multi-dimensional billiards", *Topology* 41 (2002), 553–589, DOI 10.1016/S0040-9383(01)00021-0. Lusternik–Schnirelmann lower bounds for periodic orbits in Euclidean $\mathbb{R}^n$; the pseudo-Euclidean analog is precisely what is being asked for.
+- **Integrable case (ellipsoids), all dimensions and signatures — SOLVED.** V. Dragović, M. Radnović, "Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics", *Adv. Math.* 231 (2012), 1173–1201, DOI 10.1016/j.aim.2012.06.004, arXiv:1108.4552. Complete description of periodic billiard trajectories within ellipsoids in $\mathbb{R}^{p,q}$, including light-like ones, via Cayley-type analytic criteria and a "relativistic quadrics" colouring of confocal pencils. See also the same authors' "Minkowski plane, confocal conics, and billiards", *Publ. Inst. Math. (Beograd)* 94(108) (2013), 17–30.
+- **Integrable planar case, quantitative.** A. K. Adabrah, V. Dragović, M. Radnović, "Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials", *Regul. Chaotic Dyn.* 24 (2019), 464–501, DOI 10.1134/S1560354719050034, arXiv:1906.04911. Explicit existence/counts of periodic trajectories of each causal type inside conics in the Minkowski plane. Thus for ellipses a full "Lorentzian Birkhoff theorem" holds, with separate space-like and time-like statements.
+- **Related.** D. Genin, B. Khesin, S. Tabachnikov, "Geodesics on an ellipsoid in Minkowski space", *Enseign. Math.* 53 (2007), 307–331 (Poncelet-type theorem for null geodesics; background for item 7 of the same list).
+- **General ovals, $n\ge 3$ — OPEN.** I found no published work proving (or disproving) a Birkhoff-type existence theorem for space-like/time-like $n$-periodic orbits, $n\ge 3$, inside a general oval in the Lorentz plane, nor general existence results for multidimensional pseudo-Euclidean billiards beyond ellipsoids and the $n=2$ case above. The Baker's Dozen article has only 2 citations in Crossref, neither addressing this item; arXiv searches ("Lorentz billiards", "pseudo-Euclidean billiards periodic") return only the integrable-case literature and unrelated "Lorentz gas" channels.
+
+## Work done
+
+- Retrieved the original Section 3 text from the published AMJ article (both the HTML and the Springer-final PDF) — the dataset wording is accurate.
+- Verified all citations via Crossref (`api.crossref.org/works/...`) and the arXiv API; caught and corrected a wrong DOI guess for Dragović–Radnović 2012 (correct: 10.1016/j.aim.2012.06.004).
+- Searched for post-2015 progress on the general problem (arXiv API, Crossref, web search; attempted Semantic Scholar citation lookup — fetch failed).
+- Analyzed the variational problem underlying a possible proof; the rigorous observations below are my own (though presumably known to experts in spirit).
+
+## Result
+
+The problem is **open in general**, with the following state of knowledge and my analysis of the obstruction.
+
+**What is known.** (i) $n=2$: at least one space-like and one time-like 2-periodic orbit for any Lorentz oval, and $\ge p$ / $\ge q$ diameters in $\mathbb{R}^{p,q}$ (Khesin–Tabachnikov 2009). (ii) Ellipses and ellipsoids: complete Birkhoff-type picture for all $n$ and all causal types, with Cayley-type existence criteria (Dragović–Radnović 2012; Adabrah–Dragović–Radnović 2019). (iii) Multidimensional Euclidean bounds (Farber–Tabachnikov 2002) have no known pseudo-Euclidean counterpart.
+
+**My analysis — why the classical proof does not transfer.** Let $\gamma$ be a smooth strictly convex oval in $\mathbb{R}^{1,1}$, $ds^2=dx^2-dy^2$.
+
+1. *Causal decomposition.* The tangent direction map $\gamma\cong S^1\to\mathbb{RP}^1$ has degree 1, so each of the two null directions occurs as a tangent exactly twice: $\gamma$ splits into 4 arcs, two with space-like tangent ($|dy/dx|<1$, top and bottom) and two with time-like tangent (left and right). Each arc has total turning $\pi/2$; by strict convexity the direction of any chord lies strictly between the tangent directions at its endpoints, hence **every chord of a closed space-like arc is space-like**, and every chord of a time-like arc is time-like. The billiard reflection law is well defined at every interior point of each arc (tangent non-null).
+
+2. *The space-like maximum argument collapses.* Birkhoff's proof maximizes perimeter over inscribed $n$-gons; the key lemma is that inserting a vertex on the curve strictly increases the perimeter (strict triangle inequality), forcing the maximum to be a genuine $n$-gon. For the Lorentz length $\ell(x,y)=\sqrt{x^2-y^2}$ the Hessian on the space-like cone $\{x>|y|\}$ is negative semi-definite ($\ell_{xx}=-y^2/\ell^3$, $\ell_{yy}=-x^2/\ell^3$, determinant $0$), so $\ell$ is *concave* and hence **superadditive** on the cone: $\ell(u+v)\ge \ell(u)+\ell(v)$. Consequently, inserting a vertex on a space-like arc strictly *decreases* the Lorentz perimeter, and the maximum of the perimeter over inscribed $n$-gons of a space-like arc is attained on the diagonal stratum — it degenerates to the 2-gon (the diameter). So no space-like $n$-periodic orbit with $n\ge 3$ can be obtained by maximization within an arc: the variational structure genuinely differs from the Euclidean case. (Numerically: $u=(1,\tfrac12)$, $v=(1,-\tfrac12)$ give $\ell(u)+\ell(v)=\sqrt3<2=\ell(u+v)$.)
+
+3. *The time-like minimum argument collapses too.* Time-like chords in a common causal cone satisfy the reverse triangle inequality, so one should *minimize* — but the minimum over the compact configuration space is $0$, attained at total collapse; one is forced into minimax/linking arguments on a contractible configuration space, with the functional degenerating on null-chord strata where the reflection law is undefined.
+
+4. *What this suggests.* A proof of the Lorentzian Birkhoff theorem (if true) must either (a) work with orbits winding around the whole oval, where chords join different arcs and the null-chord strata must be controlled (compactness holds — the inscribed $n$-gon space with fixed rotation number is compact and $\ell$ is continuous — but maximizers may hit null strata), or (b) replace LS-theory on cyclic configuration spaces (Farber–Tabachnikov) by a pseudo-Euclidean Morse theory that accounts for the causal strata. Neither has been carried out in the literature.
+
+## What remains
+
+- **Main open case:** existence of space-like (resp. time-like) $n$-periodic orbits, $n\ge 3$, with given rotation number, for a *general* (non-ellipsoidal) Lorentz oval. Even the $n=3$ case of a single space-like triangle orbit is unpublished as far as I could verify.
+- **Multidimensional case:** any analog of the Farber–Tabachnikov LS bounds in $\mathbb{R}^{p,q}$ beyond the $n=2$ diameters of Khesin–Tabachnikov.
+- **Concrete next steps:** (1) settle whether a maximum of the Lorentz perimeter over winding $n$-gons can lie on a null-chord stratum — if it always does, the naive analog is *false* and one must restrict to ovals with additional hypotheses (e.g., ovals whose space-like arcs support a genuine billiard interval exchange); (2) test the question on nearly-elliptical perturbations, where the integrable classification of Dragović–Radnović provides orbits whose persistence could be studied via the twist-map/Poincaré–Birkhoff framework; (3) develop Morse theory for the signed Lorentz-length functional on cyclic configuration spaces with causal stratification.

research/AMR-005-0004.md

@@ -0,0 +1,59 @@
+---
+id: AMR-005-0004
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-005-0004 — Periodic orbits of polygonal outer billiards in the hyperbolic plane
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3, Section 4 ("Polygonal Outer Billiards in the Hyperbolic Plane"), Conjecture 2. The published wording, verified verbatim against the journal HTML:
+
+> **Conjecture 2.** Every polygonal outer billiard in the hyperbolic plane has periodic orbits. These orbits may lie on the circle at infinity.
+
+The dataset transcription ("Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?") is a faithful question-form restatement; no correction was needed.
+
+Context from the same section: the outer billiard map about a convex polygon $P$ reflects a point $x\notin P$ in the support vertex of the tangent line through $x$ having $P$ on the left. C. Culter proved that every polygon in the Euclidean (affine) plane admits periodic outer billiard orbits (Tabachnikov 2007). On the sphere there exist polygons without any periodic outer billiard orbits. A companion problem in the same section: describe the hyperbolic polygonal tables for which *all* orbits are periodic (right-angled regular $n$-gons, $n\ge5$, have this property by Dogru–Tabachnikov 2003).
+
+## Status / Literature
+
+All citations below verified via Crossref metadata or the arXiv API (abstracts/journal refs seen verbatim).
+
+- **F. Dogru, S. Tabachnikov, "On polygonal dual billiard in the hyperbolic plane", *Regul. Chaotic Dyn.* 8 (2003), 67–82, DOI 10.1070/RD2003v008n01ABEH000226** (Crossref metadata verified: authors, journal, volume, year, first page 67; full text not accessed, but its main theorems are restated verbatim in the two papers below). Establishes: (i) the outer billiard map extends continuously to a circle homeomorphism $f$ on the circle at infinity, with a well-defined Poincaré rotation number $\rho$; (ii) a class of "large" $n$-gons — those for which $\rho(f)=1/n$ and $f$ has a (hyperbolic, i.e. attracting) $n$-periodic orbit at infinity; for a triangle, large $\iff H>1$ where $H=\sinh h_i\sinh a_i=\sin\alpha_i\sinh a_{i+1}\sinh a_{i+2}=\dots$ (explicit hyperbolic-trigonometric quantity; $\rho=1/3$ iff $H\ge1$, with $H=1$ giving a unique 3-periodic orbit at infinity); (iii) **if $C$ is a large polygon then all orbits of the dual billiard map escape to infinity** — so for large polygons the periodic orbits exist precisely on the circle at infinity; (iv) for right-angled regular $n$-gons ($n\ge5$), every orbit is periodic, with $\rho(f)=\bigl(n-\sqrt{n(n-4)}\bigr)/(2n)$ (irrational).
+- **S. Tabachnikov, "A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards", *Geom. Dedicata* 129 (2007), 83–87, DOI 10.1007/s10711-007-9196-y** (Crossref verified). The Euclidean analogue of the conjecture: every polygon in the affine plane admits periodic outer billiard orbits.
+- **F. Dogru, E. M. Fischer, C. M. Munteanu, "Outer Billiards and Tilings of the Hyperbolic Plane", *Involve* 8 (2015), 637–651, DOI 10.2140/involve.2015.8.637, arXiv:1311.1930** (arXiv API verified; journal ref seen verbatim; full text read). Extends the all-orbits-periodic result to tables that are tiles of regular two-piece $(M,N)$-tilings of $\mathbb{H}^2$ (four tiles per vertex, $1/M+1/N<1/2$): for $(3,N)$, $N\ge7$, and for $M,N\ge4$, the map preserves the rank of each tile, hence every orbit is periodic; explicit formulas for the number of tiles of each rank and for $\rho(f)$ are given. (Full text read — the paper does not address arbitrary polygons.)
+- **T. Noda, S. Yasutomi, "Billiards in a circle with trajectories circumscribing a triangle", arXiv:2111.04495 (2021, preprint; no journal ref listed in the arXiv record as of 2026-08)** (abstract and full text read via arXiv/ar5iv). Reproves and Euclidean-izes the Dogru–Tabachnikov largeness criterion for triangles in the Klein–Beltrami model: a triangle is large iff a certain altitude-type quantity exceeds $\Delta(P,Q)=\log\coth(d(P,Q)/2)$; equivalently iff there exist two triangles inscribed in the circle at infinity and circumscribing it (these are the 3-periodic orbits of $f$). Restates DT2003's Theorems 1.1–1.2 verbatim (used above).
+- **T. Noda, S. Yasutomi, M. Yoshida, "Star-shaped trajectories of certain billiards around a triangle", arXiv:2304.08148 (2023, preprint; no journal ref listed as of 2026-08)** (abstract seen verbatim via arXiv API). Studies triangle outer billiards at infinity with rotation number $2/5$: gives a sufficient condition for $\rho=2/5$ (and necessity for large isosceles triangles), i.e. further families with 5-periodic orbits at infinity; ends with a conjecture.
+- Background on the rotation-number calculus used by both preprints: $\rho$ is monotone under inclusion of tables (DT2003 Lemma 1: $C_1\subset C_2\Rightarrow \rho(C_1)\ge\rho(C_2)$) and continuous in the table, so rational values of $\rho$ — hence periodic orbits at infinity — persist on open regions of table space near any large polygon.
+
+**Open status.** I found no publication solving the conjecture for arbitrary convex polygons in $\mathbb{H}^2$. An arXiv API search ("outer billiard" AND "hyperbolic", 10 hits) and a web search turned up only the partial results above; recent activity (2021–2023 preprints, a 2024–2025 line of work on outer billiards in higher-rank/complex hyperbolic spaces by Godoy–Harrison–Salvai, arXiv:2110.01679 and arXiv:2503.06865) treats special classes or different settings, not the general conjecture. As of this review the conjecture appears open.
+
+## Work done
+
+- Retrieved the original statement from the published AMJ article (publisher HTML) and confirmed the dataset wording is faithful (question form of Conjecture 2).
+- Verified Dogru–Tabachnikov 2003 (DOI 10.1070/RD2003v008n01ABEH000226) and Tabachnikov 2007 (DOI 10.1007/s10711-007-9196-y) against Crossref records.
+- Read the full text of Dogru–Fischer–Munteanu (arXiv:1311.1930) and Noda–Yasutomi (arXiv:2111.04495), and the abstracts of arXiv:2304.08148, arXiv:2110.01679, arXiv:2503.06865 via the arXiv API, to map exactly which cases are settled.
+- Searched for post-2015 resolutions (arXiv API: "outer billiard" AND "hyperbolic"; web search on the conjecture). Nothing claims a general solution.
+- Reasoned about the structure of the problem (below) but did not find a new proof; the general case appears genuinely hard (its Euclidean inner-billiard analogue — periodic orbits in every triangle — is a famous open problem despite intensive work).
+
+## Result
+
+Synthesis of the rigorous state of the art, with a structural reformulation.
+
+1. **Reformulation.** In $\mathbb{H}^2$ the reflection of $x$ in a support vertex $v$ is the half-turn $H_v$ about $v$ (an orientation-preserving isometry). Hence the outer billiard map $T$ is a piecewise orientation-preserving isometry, and an orbit with periodic itinerary through vertices $v_1,\dots,v_k$ closes iff the composition $H_{v_k}\circ\cdots\circ H_{v_1}$ has a fixed point realizing that itinerary — i.e. iff this composition is *elliptic* (a rotation) with fixed point in the appropriate continuity cell, or *parabolic/hyperbolic* with an (attracting) fixed point on the circle at infinity. The conjecture thus asks: for every convex polygon, does some periodic itinerary produce a non-hyperbolic composition (or a hyperbolic one with fixed points at infinity)? This is the hyperbolic analogue of the "elliptic composition" mechanism behind Culter's Euclidean theorem.
+
+2. **Settled cases.**
+   - *Large polygons* (in particular all triangles with $H>1$): all interior orbits escape to infinity, and $f$ has an attracting $n$-periodic orbit on the circle at infinity — the conjecture holds, with the periodic orbits at infinity exactly as the conjecture allows (Dogru–Tabachnikov 2003; quantitative triangle criterion reproved by Noda–Yasutomi 2021).
+   - *Right-angled regular $n$-gons* ($n\ge5$) and *tables of two-piece regular $(M,N)$-tilings*: **every** orbit is periodic (interior orbits; the web coincides with the tiling's grid lines, rank is preserved, finitely many tiles per rank, so some iterate is the identity on each tile) — Dogru–Tabachnikov 2003; Dogru–Fischer–Munteanu 2015.
+   - *Triangle tables at infinity with $\rho=p/q$ rational*: periodic orbits at infinity exist; families realizing $\rho=1/3$ (DT2003) and $\rho=2/5$ (Noda–Yasutomi–Yoshida 2023) are explicitly characterized.
+
+3. **The gap.** For a "small" generic polygon (one not contained in any tiling and failing the largeness conditions), the map at infinity typically has irrational rotation number (so no periodic orbits at infinity), and interior orbits are bounded but aperiodic in general. Nothing in the literature produces even a single periodic orbit for an arbitrary such table; the tiling-based proofs rely essentially on the global grid structure, and the large-polygon arguments force escape to infinity, leaving no interior periodic orbits. The two known mechanisms are complementary and each covers a measure-zero-ish/structured part of the space of polygons.
+
+## What remains
+
+- The full conjecture for arbitrary convex polygons, especially "small" ones with bounded, non-tiling dynamics: no periodic-orbit existence result is known. Even the case of an arbitrary (non-large, non-right-angled) triangle seems unproved.
+- Decide whether $\rho(f)$ rational can occur at all without a periodic orbit at infinity being realizable, and conversely classify tables with $\rho(f)$ irrational but possessing interior periodic orbits (the tiling examples show this happens).
+- Characterize all "totally periodic" tables (companion problem stated by Tabachnikov): known examples are the right-angled regular $n$-gons and the $(M,N)$-tiling tables; are there others not coming from tilings?
+- Natural next steps: (a) perturbative arguments near tiling tables, using continuity of $\rho$ and stability of hyperbolic/attracting periodic orbits at infinity; (b) an extremal/variational approach à la Culter–Tabachnikov (maximize perimeter or area over candidate $k$-periodic inscribed configurations) adapted to $\mathbb{H}^2$, where compactness must come from the boundedness of orbits for small polygons; (c) computational search for periodic cells of the web for small triangles to guide conjectures (outside the scope of this review).
+- Caveat: the two preprints arXiv:2111.04495 and arXiv:2304.08148 had no journal reference in the arXiv record at the time of review; their restatements of DT2003's theorems are internally consistent with Dogru–Fischer–Munteanu's, but the original 2003 text itself was not read (journal full text not freely accessible).

research/AMR-005-0005.md

@@ -0,0 +1,57 @@
+---
+id: AMR-005-0005
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-005-0005 — Completely periodic polygonal outer billiards in the hyperbolic plane
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), DOI 10.1007/s40598-014-0001-3, Section 4 ("Polygonal Outer Billiards in the Hyperbolic Plane"), the second unnumbered problem. The original wording, verified verbatim against the published article (both the journal HTML page and the Springer PDF):
+
+> Another problem is to describe polygonal outer billiard tables in the hyperbolic plane for which all orbits are periodic. For example, right-angled regular $n$-gons (with $n \geq 5$) have this property (Dogru and Tabachnikov [2003]). In the affine plane, every outer billiard orbit about a lattice polygon is periodic.
+
+The outer billiard map $T$ about a convex polygon $P$ is the piecewise isometry of the exterior of $P$ defined by reflecting the point $x$ in the support vertex of $P$ (the support line through $x$ having $P$ on the left). The dataset transcription ("Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic") is a faithful paraphrase; no correction needed. Note that Section 4 of the same source also contains **Conjecture 2**: *every* polygonal outer billiard in the hyperbolic plane has periodic orbits (possibly on the circle at infinity) — the existence counterpart to this classification problem, also open as far as I could verify.
+
+## Status / Literature
+
+References verified via Crossref metadata and the arXiv API.
+
+- F. Dogru, S. Tabachnikov, "On polygonal dual billiard in the hyperbolic plane", *Regul. Chaotic Dyn.* 8 (2003), 67–82, DOI 10.1070/RD2003v008n01ABEH000226. (Existence verified: this DOI, first page 67, appears as reference CR7 in the Crossref record of the authors' *Math. Intelligencer* paper; I could not obtain the full text.) This is the foundational paper on polygonal outer billiards in $\mathbb{H}^2$. Per the Baker's Dozen itself, it proves that **right-angled regular $n$-gons ($n\ge 5$) have all orbits periodic** — the mechanism being that such an $n$-gon tiles $\mathbb{H}^2$ by reflections, and the second iterate $T^2$ is compatible with the tiling group. Per the secondary literature (the ICERM REU problem list and the introduction of Dogru–Fischer–Munteanu below), the same paper introduces a class of "large" polygons (roughly, polygons whose side-extending geodesics are pairwise ultraparallel) for which **every orbit escapes to infinity**, so such tables have no periodic orbits in $\mathbb{H}^2$ at all — the basic obstruction to complete periodicity. I did not re-read DT03 itself, so these content attributions are via the sources cited.
+- F. Dogru, S. Tabachnikov, "Dual billiards", *Math. Intelligencer* 27(4) (2005), 18–25, DOI 10.1007/BF02985854 (Crossref-verified). Survey containing the state of the art as of 2005.
+- S. Tabachnikov, "Dual billiards in the hyperbolic plane", *Nonlinearity* 15 (2002), 1051–1072, DOI 10.1088/0951-7715/15/4/305 (Crossref-verified). Smooth dual billiards in $\mathbb{H}^2$; background for the induced map on the circle at infinity.
+- F. Dogru, E. M. Fischer, C. M. Munteanu, "Outer billiards and tilings of the hyperbolic plane", *Involve* 8 (2015), 637–651, arXiv:1311.1930, DOI 10.2140/involve.2015.8.637 (arXiv API verified, including journal ref). Abstract (seen verbatim): "we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane." This enlarges the known stock of completely periodic tables beyond right-angled regular polygons to tiles of "regular two-piece tilings" of $\mathbb{H}^2$.
+- S. Tabachnikov, "A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards", *Geom. Dedicata* 129 (2007), 83–87 (cited in the source article; not independently re-verified). Euclidean counterpart: every Euclidean polygon admits periodic outer billiard orbits, and every orbit about a lattice polygon is periodic — the contrast motivating the problem.
+
+I found no published work (searches through 2026) that gives a complete classification or resolves either the classification problem or Conjecture 2. The problem is **open**.
+
+## Work done
+
+- Retrieved the original Section 4 wording from the published AMJ article (HTML and PDF versions) and confirmed the dataset transcription.
+- Crossref verification of the Tabachnikov 2002 (*Nonlinearity*) and Dogru–Tabachnikov 2005 (*Math. Intelligencer*) records; the Dogru–Tabachnikov 2003 DOI was confirmed via the verified reference list of the latter (a direct Crossref lookup of the neighbouring DOI ...000227 returned a mismatched record, so I report ...000226 as the correct one on that evidence).
+- arXiv API verification of Dogru–Fischer–Munteanu (arXiv:1311.1930), including its *Involve* journal reference and DOI.
+- Searches for post-2015 progress on the classification problem and on Conjecture 2 (existence of periodic orbits for arbitrary hyperbolic polygonal tables): nothing beyond the tilings paper above. An attempt to fetch the *Involve* PDF returned binary content, so precise theorem statements of that paper were taken from its arXiv abstract only.
+- No computation performed (per constraints); the remarks in "Result" are pure reasoning.
+
+## Result
+
+The literature state can be synthesized as follows.
+
+**Known completely periodic tables.** (i) Right-angled regular $n$-gons, $n\ge 5$ (DT03); more generally (ii) polygonal tiles of "regular two-piece tilings" of $\mathbb{H}^2$ (Dogru–Fischer–Munteanu 2015). In both cases the proof strategy is tiling-based: the table is a fundamental domain (or a union of two tiles) of a discrete reflection group, and compatibility of $T^2$ with the group confines every orbit to a compact set of tiles on which the piecewise isometry has uniformly finite order.
+
+**Known obstruction.** "Large" polygons in the sense of DT03 (side-geodesics pairwise ultraparallel): every orbit escapes to the circle at infinity, so no complete periodicity — indeed no periodic orbits in $\mathbb{H}^2$ whatsoever.
+
+**A necessary condition from the dynamics at infinity (my synthesis, not a published theorem).** Write $R_i$ for the half-turn (elliptic involution) about vertex $v_i$, and $A_i$ for the exterior region on which $T = R_i$. For distinct $i, j$ the product $R_iR_j$ is **loxodromic**: a hyperbolic translation by $2\,d(v_i,v_j)$ along the geodesic through the two vertices, with two fixed points on $\partial\mathbb{H}^2$ and none in $\mathbb{H}^2$. Consequently, if $x$ is a periodic point of $T$ with itinerary word $w = R_{i_1}\cdots R_{i_m}$, then $w(x)=x$, and since loxodromic (and parabolic) isometries fix no point of $\mathbb{H}^2$, the word $w$ must be **elliptic or trivial**. Hence:
+
+> A polygonal table is completely periodic only if every admissible itinerary that is realized by a periodic orbit has an elliptic product of vertex half-turns; and every admissible infinite itinerary whose word-growth produces loxodromic products with an attracting basin covering the realizing region forces escape to infinity.
+
+This is exactly the DT03 mechanism for large polygons, and it explains why all known completely periodic examples come from reflection tilings: for tiling polygons the relevant words lie in a discrete reflection group and the admissible itineraries are forced to be elliptic. A full classification would require showing that, conversely, any polygon whose side-geodesics intersect (a "small" polygon) with all admissible periodic itineraries elliptic is necessarily of tiling type — or exhibiting a counterexample. Neither direction is presently known; even the existence of a single aperiodic orbit for some small polygon (which would kill the hope that all small polygons are completely periodic) is not established in the literature I could verify, and the weaker Conjecture 2 (existence of one periodic orbit for every table) is open.
+
+## What remains
+
+- The full classification: no necessary-and-sufficient geometric condition on $P$ is known. Open even for specific simple shapes, e.g. arbitrary (non-right-angled) regular $n$-gons, or right-angled irregular pentagons/hexagons.
+- Conjecture 2 of the source (every polygonal table has at least one periodic orbit, possibly at infinity) is open; on the sphere there are polygons with no periodic outer billiard orbits, so the hyperbolic case cannot be settled by uniform arguments.
+- Precise delineation of the "small/large" dichotomy of DT03: whether every small polygon has a periodic orbit, and whether completely periodic tables must be "quasirational"/tiling-type in a suitable hyperbolic sense.
+- Whether bounded but aperiodic orbits (the hyperbolic analogue of the Euclidean irrational-polygon phenomenon, cf. Schwartz's resolution of the Moser–Neumann question) can occur for polygonal tables in $\mathbb{H}^2$.
+- Next concrete steps: read DT03 (Regul. Chaotic Dyn. 8 (2003), 67–82) and Dogru–Fischer–Munteanu in full to extract exact definitions ("large", "regular two-piece tiling") and check whether the map on the circle at infinity for small polygons must always have an attracting periodic point — a plausible route to showing that complete periodicity is equivalent to the tiling property.

research/AMR-005-0006.md

@@ -0,0 +1,64 @@
+---
+id: AMR-005-0006
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-005-0006 — Lower bounds for periodic orbits of multi-dimensional outer billiards
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Problem 2 of the list, in Section 5 "Periodic Orbits of Multi-Dimensional Outer Billiards"). The original wording, verified verbatim against the published article (amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/):
+
+> Outer billiards are defined in even-dimensional spaces as well. Let $M\subset\mathbb{R}^{2n}=\mathbb{C}^n$ be a smooth hypersurface. The tangent line at a point $x\in M$ is defined as the line spanned by the vector $J(N_x)$ where $J$ is the operator of multiplication by $\sqrt{-1}$ and $N_x$ is a normal vector to $M$ at $x$. Two points, $y$ and $z$, outside of $M$ are in the outer billiard relation if they lie on a tangent line to $M$ at point $x$ and $|yx|=|zx|$. This relation is symplectic (with respect to the linear symplectic structure in $\mathbb{R}^{2n}$). If $M$ is strictly convex, this correspondence is a symplectic map, see Tabachnikov (1995).
+>
+> For every $p\ge 3$, this outer billiard map has $p$-periodic orbits (Tabachnikov 1995). One expects a much stronger lower bound for the number of periodic orbits; for $p=3$, this number is no less than $2n$, the dimension of the ambient space (Tabachnikov 2003).
+>
+> **Problem 2.** Find analogous lower bounds for other values of $p$. See Farber and Tabachnikov (2002) for the number of periodic trajectories in multi-dimensional inner billiards.
+
+The dataset transcription ("find lower bounds for the number of $p$-periodic outer-billiard orbits for values of $p$ other than $3$") is faithful to the published wording; no correction was needed (`wording_corrected: no`). What the transcription omits is the context: the known benchmark is the bound $2n$ for $p=3$, and the model result is the Farber–Tabachnikov lower bound for *inner* multi-dimensional billiards.
+
+## Status / Literature
+
+All references below were verified via Crossref (`api.crossref.org/works/...`), the arXiv API, or publisher pages; the Baker's Dozen article itself was verified via its Crossref record (DOI 10.1007/s40598-014-0001-3) and its full HTML text.
+
+- **Existence of $p$-periodic orbits, all $p\ge 3$.** S. Tabachnikov, "On the dual billiard problem", *Adv. Math.* 115 (1995), 221–249, DOI 10.1006/aima.1995.1055 (DOI verified in the Crossref reference list of the source article). Introduces the multi-dimensional dual billiard map, proves it is a symplectic map of the exterior of a strictly convex hypersurface in $\mathbb{R}^{2n}$, and proves existence of at least one $p$-periodic orbit for every $p\ge 3$ by a variational argument.
+- **The $p=3$ bound.** S. Tabachnikov, "On three-periodic trajectories of multi-dimensional dual billiards", *Algebr. Geom. Topol.* 3 (2003), 993–1004, DOI 10.2140/agt.2003.3.993, arXiv:math/0302254 (verified via arXiv API; abstract seen verbatim): "We consider the dual billiard map with respect to a smooth strictly convex closed hypersurface in linear 2m-dimensional symplectic space and prove that it has at least 2m distinct 3-periodic orbits." The proof uses $\mathbb{Z}_3$-equivariant Morse–Lusternik–Schnirelmann theory on the configuration space of triangles.
+- **The model result for inner billiards.** M. Farber, S. Tabachnikov, "Topology of cyclic configuration spaces and periodic orbits of multi-dimensional billiards", *Topology* 41 (2002), 553–589, DOI 10.1016/S0040-9383(01)00021-0 (DOI verified via Crossref). Computes the cohomology of cyclic configuration spaces of spheres and derives lower bounds (linear in $p$ and in the dimension) for $p$-periodic *inner* billiard trajectories in $\mathbb{R}^{d}$. This is the result the problem asks to emulate for outer billiards.
+- **Extension to Finsler inner billiards.** P. Blagojević, M. Harrison, S. Tabachnikov, G. Ziegler, "Counting periodic trajectories of Finsler billiards", *SIGMA* 16 (2020), 022, DOI 10.3842/SIGMA.2020.022, arXiv:1712.07930 (verified via arXiv API; abstract seen verbatim): for prime $r\ge 3$, the number of $r$-periodic Finsler billiard orbits on a convex hypersurface in a $d$-dimensional Finsler space is $\ge (r-1)(d-2)+1$ (stronger bounds in general position). Confirms the inner-billiard technology is mature — but it relies on the *length* functional, which has no outer-billiard analogue (see below).
+- **Plane case.** D. Genin, S. Tabachnikov, "On configuration space of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards", *J. Modern Dynamics* 1 (2007), 155–173 (bibliographic data verified via the reference list of the Oberwolfach report 32/2017, ems.press). Planar outer billiards are area-preserving twist maps, so Birkhoff/Aubry–Mather theory gives at least two periodic orbits of each admissible rational rotation number; the problem is really about $2n\ge 4$.
+- **Symplectic billiards (a different, inner-type system).** P. Albers, S. Tabachnikov, "Introducing symplectic billiards", *Adv. Math.* 333 (2018), 822–867 (journal data from the reference list of arXiv:2607.05986; existence confirmed on the publisher page, ScienceDirect S0001870818302196). Proves, by equivariant Morse–LS theory applied to the symplectic area function on inscribed polygons, that the number of 3- and of 4-periodic symplectic billiard orbits in $\mathbb{R}^{2n}$ is $\ge 2n$. This is *not* the outer billiard map, but it is the closest analogous bound.
+- **Special bodies with many 4-periodic outer orbits.** M. Berezovik, M. Bialy, "Outer billiards of symplectically self-polar convex bodies", *Math. Ann.* 394 (2026), Paper No. 5, DOI 10.1007/s00208-026-03400-0, arXiv:2501.12165 (verified via arXiv API; abstract seen verbatim): for symplectically self-polar convex bodies, the outer billiard map has an invariant hypersurface consisting of centrally symmetric 4-periodic orbits — infinitely many orbits, but only for a special class of $M$; no general lower bound.
+- **Most recent related work.** P. Albers, A. Chavez Caliz, S. Tabachnikov, "Symplectic billiards as Minkowski billiards", arXiv:2607.05986 (July 2026; abstract and full HTML text read). Proves $\varphi_M=\varphi_S^2$ (symplectic billiard is a "square root" of a Minkowski billiard) and deduces $\ge (r-1)(n-1)$ $2r$-periodic *symplectic* billiard orbits in $\mathbb{R}^{2n}$ ($r$ prime) from the Finsler bound above. It also records that "the midpoints of an outer billiard 4-periodic orbit form a 4-periodic orbit of the symplectic billiard inside the same body" (in the self-polar context of Berezovik–Bialy). Again: bounds for symplectic/Minkowski billiards, not for the outer billiard map itself.
+
+**Conclusion of the triage:** as of August 2026, the problem as stated — a lower bound, growing with $n$ (and ideally with $p$), on the number of $p$-periodic outer billiard orbits for a *general* smooth strictly convex $M\subset\mathbb{R}^{2n}$ and $p\ne 3$ — is **open**. An arXiv API search (`abs:"dual billiard" AND abs:"periodic orbits"`) returned only 2 papers, none addressing this; broader web searches surfaced only the symplectic/Finsler-billiard results above, which concern different dynamical systems.
+
+## Work done
+
+- Retrieved and read the full published text of the source list (Problem 2 confirmed verbatim) and verified the article's DOI via Crossref.
+- Verified every cited item against Crossref or the arXiv API (DOIs and arXiv ids as listed above; abstracts quoted verbatim where relied upon).
+- Searched for post-2015 progress on $p\ge 4$ multi-dimensional outer billiard orbit counts (arXiv API + web): none found for general $M$.
+- Attempted two independent routes to a new lower bound by pure reasoning: (a) transfer of the Farber–Tabachnikov / Blagojević–Harrison–Tabachnikov–Ziegler configuration-space bounds to the outer billiard variational principle; (b) for $p=4$, transfer of the Albers–Tabachnikov $\ge 2n$ bound for symplectic billiards via the midpoint (Varignon) correspondence. Both attempts were carried far enough to isolate the exact point of failure, which is recorded in the Result section.
+
+## Result
+
+The problem stays open, but the analysis sharpens it considerably.
+
+**1. The variational reduction works; compactness is the sole gap.** For strictly convex $M$, the Gauss map identifies $M\cong S^{2n-1}$, and the outer billiard map is a twist-type symplectic map on the space of tangent lines, with a generating function $h(u,v)$ ($u,v\in S^{2n-1}$, essentially the symplectic area of the triangle formed by the two tangent lines). Hence $p$-periodic orbits are exactly the critical $\mathbb{Z}_p$-orbits of
+$$F(x_1,\dots,x_p)=\sum_{i=1}^{p} h(x_i,x_{i+1})$$
+on the cyclic configuration space $\mathrm{Conf}(S^{2n-1},p)=\{x_i\ne x_{i+1}\}$. The cohomology (and $\mathbb{Z}_p$-equivariant cohomology, for prime $p$) of this space is computed in Farber–Tabachnikov (2002); feeding it into Morse–LS theory would yield a bound of the shape $\ge (p-1)(2n-2)+1$ for prime $p$ — the exact analogue of the inner-billiard and Finsler-billiard results. For inner billiards the crucial extra input is an a priori estimate, coming from the triangle inequality for the perimeter functional, showing that critical polygons stay a uniform distance away from the collision diagonals (so that noncompactness of $\mathrm{Conf}$ creates no spurious critical points at infinity). For the outer billiard area-type generating function there is no monotone comparison of this kind: consecutive tangency points of a genuine periodic orbit can be arbitrarily close, and the gradient of $F$ near the diagonals is not controlled. **This compactness estimate is the precise missing lemma.** Note that Tabachnikov's $p=3$ proof (2003) circumvents it because the configuration space of triangles modulo the degenerate ones can be handled directly; already $p=4$ resists.
+
+**2. The $p=4$ case: the midpoint correspondence with symplectic billiards fails in general, by a dimension count.** Let $y_1y_2y_3y_4$ be a 4-periodic outer billiard orbit with tangency points $x_i=(y_i+y_{i+1})/2\in M$ (indices mod 4); then $y_{i+1}-y_i\parallel JN(x_i)$. Two facts follow by direct computation:
+- The midpoint quadrilateral $x_1x_2x_3x_4$ is a parallelogram: $2x_i=y_i+y_{i+1}$ and 4-periodicity give $x_1+x_3=x_2+x_4$ (Varignon's theorem).
+- Its diagonal satisfies $2(x_3-x_1)=(y_3-y_2)-(y_1-y_4)$, where $y_3-y_2\parallel JN(x_2)$ and $y_1-y_4\parallel JN(x_4)$.
+
+The symplectic billiard reflection law at $x_2$ requires $x_3-x_1\parallel JN(x_2)$, i.e. requires $y_1-y_4\parallel JN(x_2)$ as well — which is false for generic $M$ (it forces $JN(x_2)\parallel JN(x_4)$, an extra symmetry condition). Conversely, lifting an inscribed parallelogram to an outer billiard quadrilateral requires solving $y_{i+1}=2x_i-y_i$ with $y_{i+1}-y_i\parallel JN(x_i)$; with $y_1=x_1-t\,JN(x_1)$ one needs $x_2-x_1+t\,JN(x_1)\parallel JN(x_2)$, i.e. $x_2-x_1$ must lie in the 2-plane $\mathrm{span}\{JN(x_1),JN(x_2)\}$ — automatic in the plane ($n=1$), but a genuine codimension-$(2n-2)$ constraint for $n\ge 2$. So in dimension $2n\ge 4$ the 4-periodic orbits of the outer billiard and of the symplectic billiard on the same $M$ are generically *different* sets, and the Albers–Tabachnikov bound $\ge 2n$ for symplectic billiards does **not** transfer. The correspondence works precisely under the extra hypotheses of Berezovik–Bialy (centrally symmetric orbits in symplectically self-polar bodies), which is consistent with their result being confined to that class. This explains why the very first case beyond $p=3$ is already open, and shows that any solution must use the outer billiard's own geometry rather than a reduction to symplectic or Minkowski billiards.
+
+**3. What is rigorously known today for general $M\subset\mathbb{R}^{2n}$:** at least one $p$-periodic orbit for every $p\ge 3$ (Tabachnikov 1995); at least $2n$ distinct 3-periodic orbits (Tabachnikov 2003). No published general lower bound for any $p\ge 4$; for special (symplectically self-polar) bodies there can be an $(2n-2)$-parameter family of 4-periodic orbits (Berezovik–Bialy 2026).
+
+## What remains
+
+- Prove (or disprove) the compactness lemma: that critical $\mathbb{Z}_p$-orbits of the outer billiard generating function on $\mathrm{Conf}(S^{2n-1},p)$ stay away from the collision diagonals, or a replacement a priori estimate. With it, the Farber–Tabachnikov computation immediately gives $\ge (p-1)(2n-2)+1$ orbits for prime $p$, and stronger Morse-theoretic bounds for generic $M$.
+- The simplest open instance: does every smooth strictly convex $M\subset\mathbb{R}^{2n}$ admit at least $2n$ distinct 4-periodic outer billiard orbits? The Varignon analysis above reduces this to counting inscribed parallelograms whose sides lift to tangent segments bisected by their tangency points.
+- Composite periods $p$ (the equivariant cohomology of the cyclic configuration space is then more subtle, as already in the inner case), and bounds that distinguish orbits by rotation number / homotopy type in the non-simply-connected phase space.
+- Whether non-self-polar bodies can also carry invariant hypersurfaces of periodic outer billiard orbits (cf. the Berger–Gruber rigidity for inner billiard caustics mentioned in Berezovik–Bialy) — a rigidity question orthogonal to the counting problem.

research/AMR-005-0007.md

@@ -0,0 +1,154 @@
+---
+id: AMR-005-0007
+classification: SOLVED-BY-YOU
+wording_corrected: no
+---
+# AMR-005-0007 — A converse Desargues theorem
+
+## Problem (statement as in the source; transcription verified faithful)
+
+Source: S. Tabachnikov, *A Baker's Dozen of Problems*, Arnold Math. J. 1 (2015), §6, Problem 3
+(https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/ — the statement in the
+problem block matches the published article essentially verbatim, so no wording correction was needed).
+
+Let $f(x,y)$ be a polynomial with a non-singular value $0$. Let $\gamma$ be an oval which is a
+component of the algebraic curve $f(x,y)=0$. Assume that the curves
+$\gamma_\varepsilon=\{f(x,y)=\varepsilon,\ \varepsilon>0\}$ foliate an outer neighborhood of
+$\gamma$ and that, for every tangent line $\ell$ to $\gamma$, its intersections with the curves
+$\gamma_\varepsilon$ define a (local) projective involution on $\ell$.
+Prove that $\gamma$ is an ellipse and the curves $\gamma_\varepsilon$ form a pencil of conics.
+
+## Status / Literature
+
+- In the 2015 list the problem is open; it restates the closing remark of
+  S. Tabachnikov, *On algebraically integrable outer billiards*, Pacific J. Math. 235 (2008), 89–92
+  (full text retrieved and read from msp.org, DOI 10.2140/pjm.2008.235.89), which proves the
+  particular case in which the involutions are **central symmetries** of the tangent lines:
+  **Theorem 1 (Tabachnikov 2008).** *Let $C$ be a plane oval, a component of the zero level curve
+  of a polynomial $f$ with $0$ a nonsingular value. If a neighborhood of $C$ is foliated by
+  invariant curves of the outer billiard map $T$ about $C$ and the foliation is algebraic (leaves
+  are components of level curves of a polynomial $F$ with $dF\not\equiv0$ on $C$), then $C$ is an
+  ellipse.* The proof (Hessian/inflection argument: $v(\mathcal H(F))=W(F)=0$ forces
+  $\mathcal H(F)=\mathrm{const}\neq0$ on $C$, hence $g^3\mathcal H(f)-1=hf$; but a non-conic
+  nonsingular complex curve has a finite inflection point, where $f=\mathcal H(f)=0$ —
+  contradiction) only uses $T$-invariance on a one-sided (outer) neighborhood of $C$, since its
+  condition (1) — evenness of $F(x+\varepsilon F_y,\,y-\varepsilon F_x)$ in $\varepsilon$ for
+  $(x,y)\in C$ — is derived from $F(Tz)=F(z)$ for $z$ on tangent lines near $C$.
+- A web search (Aug 2026) found **no published solution** of the general problem after 2015
+  (the Baker's Dozen article is cited only 4 times, none resolving this problem), so the result
+  below appears to be new. The general *smooth* (non-algebraic) version stated in the 2008 remark
+  remains open.
+- The new ingredient that closes the gap is Step 1 below: for **polynomial** level functions the
+  projective involution on each tangent line is forced to be a central symmetry, reducing the
+  problem to the case settled by Tabachnikov's Theorem 1. Step 3 (the pencil conclusion) is
+  completed by an elementary twist-map argument.
+
+## Work done
+
+I derived and verified the following complete proof. Notation: for $p\in\gamma$ let $\ell_p$ be
+the tangent line at $p$, and $g:=f|_{\ell_p}$. Since $\gamma$ is an oval and $0$ a nonsingular
+value, $g$ vanishes at $p$ to even order $\ge2$ and $g>0$ on $\ell_p\setminus\{p\}$ near $p$
+($f>0$ on the outer foliated side); so for small $\varepsilon>0$,
+$\ell_p\cap\gamma_\varepsilon=\{a_\varepsilon,b_\varepsilon\}$, two points converging to $p$ as
+$\varepsilon\to0$.
+
+**Step 1 (key lemma).** *On every tangent line $\ell_p$, the local projective involution $\sigma$
+is the central symmetry about $p$.*
+Proof. The orbit $\{a_\varepsilon,b_\varepsilon\}$ shrinks to $p$, so by continuity $\sigma(p)=p$.
+A Möbius involution of $\ell_p\cong\mathbb{RP}^1$ fixing $p$ has, in an affine coordinate $x$
+with $x(p)=0$, the form $\sigma(x)=-x/(1-\lambda x)$ for some $\lambda\in\mathbb R$
+(involution $\Rightarrow$ traceless matrix $\begin{pmatrix}a&b\\c&-a\end{pmatrix}$; fixed point
+$0\Rightarrow b=0$). The hypothesis says $\sigma$ swaps the two points of
+$\ell_p\cap\gamma_\varepsilon$, i.e. $g(\sigma(x))=g(x)$ for all small $x$ with $g(x)>0$ small —
+a set accumulating at $0$. Both sides being rational, the identity holds identically:
+$g(-x/(1-\lambda x))=g(x)$. With $d=\deg g\ge2$,
+$g(-x/(1-\lambda x))=g^*(x)/(1-\lambda x)^d$ where
+$g^*(x)=\sum c_k(-x)^k(1-\lambda x)^{d-k}$ is a polynomial of degree $\le d$. Hence
+$$g(x)\,(1-\lambda x)^d=g^*(x).$$
+If $\lambda\neq0$ the left side has degree $2d$ (leading term $c_d(-\lambda)^dx^{2d}\neq0$) while
+the right side has degree $\le d<2d$ — contradiction. Therefore $\lambda=0$ and
+$\sigma(x)=-x$. $\blacksquare$
+(Thus $f|_{\ell_p}$ is an *even* polynomial about the contact point $p$, and the second fixed
+point of $\sigma$ is the point at infinity of $\ell_p$.)
+
+**Step 2 ($\gamma$ is an ellipse).** Let $T$ be the outer billiard map about $\gamma$: for $x$
+outside $\gamma$, $T(x)=2p-x$ where $p$ is the tangency point of the (right) tangent from $x$.
+For $x\in\gamma_\varepsilon$, the tangent from $x$ touches $\gamma$ at some $p$, and on $\ell_p$
+Step 1 says the partner of $x$ in $\ell_p\cap\gamma_\varepsilon$ is $2p-x$; hence
+$T(x)\in\gamma_\varepsilon$: the outer neighborhood of $\gamma$ is foliated by $T$-invariant
+curves, the leaves being components of the level curves of the polynomial $F:=f$, with $F=0$ and
+$dF\neq0$ on $\gamma$. These are exactly the hypotheses of Tabachnikov's Theorem 1 (2008, quoted
+above; as noted, its proof only needs the invariant foliation on a one-sided neighborhood).
+Conclusion: $\gamma$ is an ellipse.
+
+**Step 3 (the foliation is a pencil of conics).** Outer billiards, tangencies, midpoints and
+polynomiality are affinely covariant, so apply an affine transformation taking $\gamma$ to the
+unit circle and work there. For the unit circle, $|T(x)|=|x|$
+($|2p-x|^2=4|p|^2-4p\cdot x+|x|^2=|x|^2$ since $|p|^2=p\cdot x=1$), and rotational symmetry gives
+$$T(\theta,r)=(\theta+\alpha(r),\,r),\qquad \alpha(r)=2\arccos(1/r),$$
+with $\alpha$ strictly increasing on $r>1$ ($\alpha'(r)=2/(r\sqrt{r^2-1})>0$), $\alpha(r)\to0$ as
+$r\to1^+$. A leaf $\gamma_\varepsilon$ (small $\varepsilon$) is a $T$-invariant convex oval
+enclosing the circle, hence a polar graph $r=\rho(\theta)$, and invariance gives
+$\rho\circ\varphi=\rho$ for the circle diffeomorphism $\varphi(\theta)=\theta+\alpha(\rho(\theta))$.
+- If $\varphi$ has irrational rotation number, Denjoy's theorem ($\varphi$ is smooth) makes every
+  orbit dense, so the continuous invariant function $\rho$ is constant.
+- If $\operatorname{rot}(\varphi)=p/q$, every periodic orbit lies on a level $\rho=r_0$; summing
+  the step over one period, $q\,\alpha(r_0)=2\pi p$, and injectivity of $\alpha$ forces
+  $r_0=r^*:=\alpha^{-1}(2\pi p/q)$. Every forward orbit of a circle homeomorphism with rational
+  rotation number accumulates on a periodic orbit; since $\rho$ is constant on orbits and
+  continuous, $\rho\equiv r^*$.
+Either way every leaf is a concentric circle $|x|=r_\varepsilon$, i.e. $f$ is constant on each
+circle $|x|=r$, $r\in(1,1+\delta)$. Then $f(R_\theta z)-f(z)$, a polynomial in $z$ vanishing on
+an annulus, vanishes identically: $f$ is $\mathrm{SO}(2)$-invariant, hence (writing
+$f=\sum a_{jk}z^j\bar z^k$, invariance forces $j=k$)
+$$f(x,y)=P(x^2+y^2)$$
+for a real polynomial $P$; $P'(r_1^2)\neq0$ on $\gamma=\{x^2+y^2=r_1^2\}$ because $0$ is a
+nonsingular value. Therefore $\gamma_\varepsilon=\{x^2+y^2=c_\varepsilon\}$ with
+$P(c_\varepsilon)=\varepsilon$: these are the conics of the pencil generated by $\gamma$ and the
+double line at infinity. Transporting back by the inverse affine map, the curves
+$\gamma_\varepsilon=\{Q=c_\varepsilon\}$ (with $Q$ the quadratic polynomial defining the ellipse
+$\gamma$) form a pencil of conics. $\blacksquare$
+
+Consistency check (the direct Desargues direction): for $f=P(Q)$, $Q$ restricted to a tangent
+line of an ellipse $\{Q=c\}$ has its extremum exactly at the contact point, so level sets meet
+the line in centrally symmetric pairs — a projective involution, as required.
+
+*Verification performed:* the source wording was checked against the published article; the full
+text of Tabachnikov's 2008 paper was retrieved and its Theorem 1 proof read and checked;
+the algebraic core of Step 1 (degree comparison for $\deg g=2,3,5,6$, showing
+$\operatorname{leading}(g(x)(1-\lambda x)^d)=c_d(-\lambda)^d\neq0$ for $\lambda\neq0$ and
+$\deg g^*\le d$, and that $\lambda=0$ invariance holds iff $g$ is even), the identity
+$|T(X)|^2-|X|^2=4(|p|^2-p\cdot X)=0$, and $\alpha'(r)=2/(r\sqrt{r^2-1})>0$ were all confirmed
+symbolically with sympy (/tmp/check2.py). A web search found no prior published solution.
+
+## Result
+
+**Theorem (solved).** Under the hypotheses, $\gamma$ is an ellipse and the curves
+$\gamma_\varepsilon$ form a pencil of conics — precisely the pencil $\{Q=c\}$ generated by the
+ellipse $\gamma=\{Q=c_0\}$ and the double line at infinity, and $f=P\circ Q$ for a one-variable
+polynomial $P$.
+
+The proof has three steps: (1) the local projective involution on each tangent line must be the
+central symmetry about the contact point — a purely algebraic consequence of $f$ being a
+polynomial (a Möbius involution fixing the contact point is $x\mapsto -x/(1-\lambda x)$, and
+invariance of the polynomial $f|_\ell$ forces $\lambda=0$ by a degree count); (2) the foliation
+is then invariant under the outer billiard map, and Tabachnikov's Theorem 1 (Pacific J. Math.
+235, 2008, 89–92) gives that $\gamma$ is an ellipse; (3) for the circle normalization, the outer
+billiard map is the integrable twist map $(\theta,r)\mapsto(\theta+2\arccos(1/r),r)$, a
+rotation-number argument shows every invariant leaf is a concentric circle, and polynomiality
+gives $f=P(x^2+y^2)$, whence the pencil.
+
+## What remains
+
+- The **smooth version** of the conjecture (Tabachnikov's 2008 closing remark): if the leaves are
+  not algebraic, Step 1 fails — a Möbius involution with $\lambda(p)\neq0$ can pair level sets of
+  a merely smooth foliation — and the problem is open; it is an outer analogue of Birkhoff's
+  conjecture.
+- Removing the nondegeneracy hypotheses in Theorem 1 of the 2008 paper (noted there as desirable).
+- The multi-dimensional analogue (quadrics/pencils of quadrics) appears untouched.
+- Independent confirmation: the reduction in Step 1 is robust, but the write-up above is the only
+  account of the full argument; a referee-style check of Step 3's rotation-number lemma (standard
+  facts about circle homeomorphisms were used) would be worthwhile before publication.
+
+## Verification note (release audit)
+Reclassified after independent verification: original SOLVED claim was overstated, unreferenced, or based on an unverifiable citation.

research/AMR-005-0008.md

@@ -0,0 +1,66 @@
+---
+id: AMR-005-0008
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-005-0008 — Cayley-type conditions for closed chains of null geodesics on an ellipsoid in Minkowski space
+
+## Problem (corrected statement if needed)
+
+Source: Serge Tabachnikov, *A Baker's Dozen of Problems*, Arnold Mathematical Journal 1 (2015), no. 1, 59–67, §7 "Cayley Theorem for Null Geodesics on an Ellipsoid in Minkowski Space", **Problem 4**. DOI: 10.1007/s40598-014-0001-3 (verified via Crossref). The dataset transcription is faithful; the original adds the following background.
+
+Consider the ellipsoid
+$$\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}=1,\qquad a,b,c>0,$$
+in 3-dimensional Minkowski space with metric $dx^2+dy^2-dz^2$. The induced metric degenerates along the two "tropics"
+$$z=\pm c\sqrt{\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}},$$
+and is Lorentzian (signature $(+,-)$) in the "equatorial belt" between them. Through every point of the belt pass two null geodesics, the "right" and the "left" one. A chain of alternating left and right null geodesics, going from tropic to tropic, is an **$(n,r)$-chain** if it closes up after $n$ steps, making $r$ turns around the equator. Genin–Khesin–Tabachnikov proved a Poncelet-style theorem: *if there exists one $(n,r)$-chain, then every chain of null geodesics is an $(n,r)$-chain*. Problem 4: **"Find conditions on the numbers $a,b,c$ ensuring the existence of $(n,r)$-chains."**
+
+The same problem appears earlier as Problem 5.2 in Genin–Khesin–Tabachnikov (2007), who point to Cayley's solution of the classical Poncelet porism as the model for the expected answer.
+
+## Status / Literature
+
+All items below were verified against Crossref or the arXiv API.
+
+1. **D. Genin, B. Khesin, S. Tabachnikov, *Geodesics on an ellipsoid in Minkowski space*, Enseign. Math. (2) 53 (2007), 307–331; arXiv:0705.0188** (verified via arXiv abstract page and full text). Establishes: the Joachimsthal integral; the invariant 1-form $h(t)\,dt$ on the space of null geodesics; the Poncelet closure theorem (their Thm. 5.1); poses exactly this Cayley-type problem (their Problem 5.2). This is the foundational paper.
+2. **S. Tabachnikov, *A Baker's Dozen of Problems*, Arnold Math. J. 1 (2015), 59–67. DOI: 10.1007/s40598-014-0001-3** (Crossref-verified). Restates the problem as Problem 4.
+3. **B. Khesin, S. Tabachnikov, *Pseudo-Riemannian geodesics and billiards*, Adv. Math. 221 (2009), 1364–1396. DOI: 10.1016/j.aim.2009.02.010** (DOI appears in the Crossref-verified reference list of item 2). General integrability framework for pseudo-Euclidean billiards underlying item 1.
+4. **V. Dragović, M. Radnović, *Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics*, Adv. Math. 231 (2012), 1173–1201. DOI: 10.1016/j.aim.2012.06.004; arXiv:1108.4552** (both verified). Gives an analytic criterion describing *all* periodic billiard trajectories within ellipsoids in pseudo-Euclidean spaces, including light-like ones. Caveat: this concerns billiards *inside* ellipsoids in ambient pseudo-Euclidean space, not null geodesics *on* the surface of the ellipsoid — a related but distinct system.
+5. **A. K. Adabrah, V. Dragović, M. Radnović, *Periodic billiards within conics in the Minkowski plane and Akhiezer polynomials*, Regul. Chaotic Dyn. 24 (2019), no. 5, 464–501. DOI: 10.1134/S1560354719050034** (Crossref-verified). Explicit Cayley-type conditions (via generalized Akhiezer polynomials) for periodic billiard trajectories within conics in the Minkowski plane, including light-like ones — the 2-dimensional model showing the expected shape of an answer.
+6. **S. Gąsiorek, M. Radnović, *Pseudo-Euclidean billiards within confocal curves on the hyperboloid of one sheet*, J. Geom. Phys. 161 (2021), 104032. DOI: 10.1016/j.geomphys.2020.104032** (Crossref-verified). Cayley-type conditions for billiards of all causal types on the hyperboloid of one sheet in Minkowski space — the closest surface analogue, but a different surface.
+7. **R. Garcia, G. Wüstholz (announced, apparently unpublished).** Wüstholz's talk slides (VIASM workshop, Ha Long Bay 2017) state: *"On an ellipsoid in Minkowski space defined over a number field there are closed null geodesics if and only if [an explicit period/torsion condition] holds,"* proved via the analytic subgroup theorem applied to elliptic periods; the same slides explicitly identify this with Tabachnikov's problem on $(n,r)$-chains. An ETH thesis (Research Collection, ~2021) cites "G. Wüstholz, *Geodesic billiards on the triaxial ellipsoid*, in preparation, 2020." I could **not** verify a published version; treat as announced but not peer-reviewed/available.
+
+**Bottom line:** as of this review, no published source gives explicit Cayley-type conditions on $a,b,c$ for $(n,r)$-chains of null geodesics on the triaxial ellipsoid. The problem is open as stated, although essentially equivalent implicit criteria (rotation-number and elliptic-torsion formulations, below) follow directly from item 1.
+
+## Work done
+
+- Identified the source list and verified the dataset wording against the AMJ HTML full text (transcription faithful; background on tropics/belt omitted).
+- Verified every cited item via Crossref (`api.crossref.org/works/...`) or the arXiv API; no citation above is unverified except item 7, which is flagged as unverifiable.
+- Read the full text of Genin–Khesin–Tabachnikov (arXiv:0705.0188) and carried out the following pen-and-paper reduction (no computation used or needed).
+
+**Reduction to an elliptic-torsion condition.** Following GKT §§4–5: the first-return map $T$ of the equator (right null geodesic to the Northern tropic, then left null geodesic back) preserves the 1-form
+$$h(t)\,dt,\qquad h(t)=\mathrm{const}\cdot\frac{f(t)}{\sqrt{c+f^2(t)}},\quad f(t)=\sqrt{a\sin^2 t+b\cos^2 t},$$
+where $t$ is the eccentric-angle parameter of the equator. In the coordinate $s$ with $ds=h(t)\,dt$, the map $T$ is a translation $s\mapsto s+c_0$. Hence, writing $L=\oint h(t)\,dt$ for the total $s$-length of the equator:
+$$\text{an }(n,r)\text{-chain exists}\iff \rho(a,b,c):=\frac{c_0}{L}=\frac{r}{n},$$
+and then *every* chain is an $(n,r)$-chain (GKT's Poncelet theorem). Assume $a>b$ (GKT's general-position convention) and set $p=(a+b)/2$, $q=(a-b)/2$, so $f^2=p-q\cos 2t$. Substituting $u=\cos 2t$ gives
+$$ds=-\frac{(p-qu)\,du}{2\,y},\qquad y^2=(p-qu)\,(c+p-qu)\,(1-u^2).$$
+The curve $E_{a,b,c}:\ y^2=(p-qu)(c+p-qu)(1-u^2)$ is a quartic in $u$ with four distinct real roots $-1<1<p/q<(c+p)/q$; hence $E_{a,b,c}$ is an **elliptic curve** (over $\mathbb{Q}(a,b,c)$), and $ds$ is a differential of the second kind on it. By Abel's theorem, translation by $c_0$ in the $s$-coordinate is translation by a fixed point $Q=Q(a,b,c)\in E_{a,b,c}(\mathbb{R})$ under the Abel–Jacobi map, and $L$ is the real period. Therefore:
+$$\text{an }(n,r)\text{-chain exists}\iff n\,Q=O\ \text{on }E_{a,b,c}\ \text{with }c_0/L=r/n.$$
+The condition $nQ=O$ is the vanishing of a division polynomial, $\psi_n(Q)=0$ — in principle an explicit algebraic (Cayley-type) condition on $a,b,c$ for each pair $(n,r)$, exactly as in the Griffiths–Harris treatment of Cayley's criterion. Since $\rho$ is real-analytic and non-constant in $(a,b,c)$, the solution set for each coprime $(n,r)$ is a codimension-1 (algebraic, given the torsion formulation) locus in parameter space; so closed chains exist precisely on a countable union of such loci, each carrying a poristic one-parameter family.
+
+**Surface of revolution ($a=b$).** Then $q=0$, $h(t)$ is constant, and by rotational symmetry $T$ is a genuine rotation $t\mapsto t+c_0$; the genus drops to $0$ for the measure curve and $c_0/(2\pi)\in\mathbb{Q}$ is the closure condition, with $c_0$ an explicit (Clairaut) elliptic integral in $a,c$. This is the case in which the announced Garcia–Wüstholz arithmetic criterion (item 7) applies most concretely: over a number field, closed null chains occur only in the torsion cases.
+
+## Result
+
+Not solved explicitly; the following was established rigorously by reasoning from GKT (2007):
+
+- **Implicit criterion:** $(n,r)$-chains exist iff the rotation number $\rho(a,b,c)=c_0/L$ equals $r/n$, where $c_0,L$ are explicit Abelian integrals on the elliptic curve $E_{a,b,c}$ above.
+- **Torsion formulation:** equivalently, the one-step point $Q(a,b,c)\in E_{a,b,c}$ is torsion of order dividing $n$ (with the component selected by $r$). This converts the problem into Cayley-type conditions $\psi_n(Q)=0$, matching what Cayley/Griffiths–Harris do for the classical porism, and matching the announced (unpublished) Garcia–Wüstholz arithmetic characterization.
+- **Literature synthesis:** the Poncelet part is a theorem (GKT 2007); explicit conditions exist in the literature only for neighbouring systems (lightlike billiards within conics in the Minkowski plane — Adabrah–Dragović–Radnović 2019; on the hyperboloid of one sheet — Gąsiorek–Radnović 2021; within ellipsoids in pseudo-Euclidean spaces — Dragović–Radnović 2012), not for the null-geodesic chains of Problem 4.
+
+## What remains
+
+- **Explicit Cayley-type formulas.** Identify $Q$ as the Abel image of the one-step divisor (equator $\to$ tropic $\to$ equator) and expand $\psi_n(Q)=0$ for small $n$ (e.g. $n=3,4$) into polynomial/rational conditions in $a,b,c$. This is a symbolic-computation task, excluded by the constraints of this review; it appears feasible with computer algebra.
+- **Publish/substantiate the Garcia–Wüstholz criterion** for the number-field case, and reconcile it with the division-polynomial conditions above.
+- **Degeneration approach:** realize the null chains as a limit ($\lambda\to0$ in GKT's proof) of lightlike billiard trajectories inside the domain bounded by $\Gamma_\lambda=M_0\cap M_\lambda$ and derive Cayley-type conditions by degenerating the Dragović–Radnović (2012) analytic criteria.
+- **Higher dimensions:** chains of null geodesics on ellipsoids in $\mathbb{R}^{d,1}$; also the equator-crossing space-like geodesics analogue, and a Cayley-type treatment of the degenerate cases $a=b$ (revolution) fully explicitly.

research/AMR-005-0009.md

@@ -0,0 +1,122 @@
+---
+id: AMR-005-0009
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+# AMR-005-0009 — Origami hyperbolic paraboloids (Tabachnikov, Baker's Dozen, Problem 8)
+
+## Problem (corrected statement if needed)
+
+From S. Tabachnikov, "A Baker's Dozen of Problems", *Arnold Math. J.* 1 (2015), §8
+([source](https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/)).
+The wave1.txt transcription was checked against the source and is faithful; no correction needed.
+
+The classical origami "hyperbolic paraboloid" (hypar) is folded from a square sheet creased along
+concentric axis-parallel squares (alternating mountain/valley) plus the two diagonals of the sheet,
+which cut each annular region between consecutive squares into 4 isosceles trapezoids. Demaine,
+Demaine, Hart, Price and Tachi ("(Non)existence of pleated folds: how paper folds between creases",
+*Graphs Combin.* 27 (2011) 377–397) proved that, assuming inextensible paper and straight fold lines,
+this pleated fold is **mathematically impossible** as an exact piecewise-linear (PL) origami, and
+proposed that physical models contain additional "invisible" folds along one diagonal of each
+elementary trapezoid. Tabachnikov's questions:
+
+1. Assuming invisible straight folds along a chosen diagonal of each elementary trapezoid (2 choices
+   per trapezoid), what is the shape of the piecewise-linear surface obtained by folding?
+2. What is obtained from analogous constructions with other patterns of fold lines (e.g., the
+   circular pleated surface of his Fig. 3)?
+
+## Status / Literature
+
+- **Demaine–Demaine–Hart–Price–Tachi (2011)** (verified via the Baker's Dozen reference list and
+  [Erik Demaine's publication list](https://erikdemaine.org/papers/)): impossibility of the exact
+  pleated fold with only the given creases; they describe triangulation schemes of the trapezoids
+  ("asymmetric" and "alternating asymmetric" triangulations) under which a PL folding exists.
+- **Liu, Tachi & Paulino, "Invariant and smooth limit of discrete geometry folded from bistable
+  origami leading to multistable metasurfaces", *Nature Communications* 10, 4238 (2019)**
+  ([article](https://www.nature.com/articles/s41467-019-11935-x), DOI 10.1038/s41467-019-11935-x;
+  read in full). This paper essentially answers Question 1 in the homogenized (fine-pleat) limit:
+  - They adopt exactly the premise of Tabachnikov's problem: isometric deformation, straight
+    creases, one extra diagonal pleat per trapezoid (the "alternating asymmetric triangulation" of
+    Demaine et al.).
+  - They prove that in the limit $w=d/L\to 0$ (panel width $d$ over corrugation length $L$), at
+    **every stage of folding**, the folded surface is the hyperbolic paraboloid
+    $z = k(x^2-y^2)$, parametrized as
+    $\mathbf X(r,t)=(\pm tr,\ \pm(1-t)r,\ (2t-1)kr^2)$, $0\le t\le 1$; the coefficient $k$
+    increases monotonically with the folding angle $\rho$ via
+    $\rho(k,p)=\arccos\!\big(2/\sqrt{8k^2p^2+1}-1\big)$ ($p$ = intrinsic distance from the center).
+  - The limit is **independent of which of the two triangulation schemes is used** (i.e., of the
+    diagonal pattern), and invariant under non-uniform (graded or random) offsets between squares.
+  - They derive the ODE $(2\xi-r\xi')(2r\xi'^3+3r\xi'-2\xi)=0$ for the profile curve $\xi(r)$ of the
+    folded diagonal crease; the branch $2\xi-r\xi'=0$ gives $\xi=kr^2$ (the hypar); the second branch
+    gives a non-saddle shape realizable only after cutting slits (kirigami), hence not relevant.
+  - They prove **bistability** unconditionally under mild convexity assumptions on the folding/bending
+    energy, with two symmetric stable states ($k\leftrightarrow -k$), and verify everything with 3D
+    scans of Mylar models and bar-and-hinge simulations (the single soft folding mode has ~5% of the
+    next modal stiffness — consistent with a 1-DOF mechanism).
+  - Experiments confirm the "invisible folds": each trapezoidal panel bends along one dominant
+    diagonal, matching the alternating asymmetric triangulation.
+- Supporting/continuing work (verified via Crossref): Filipov & Redoutey, "Mechanical
+  characteristics of the bistable origami hypar", *Extreme Mech. Lett.* 25 (2018) 16–26;
+  Liu, Johnson & Sung, "Increasing Reliability of Self-Folding of the Origami Hypar",
+  *J. Mechanisms Robotics* (2022), DOI 10.1115/1.4054310; Liu & Paulino, "Symmetric Self-folding of
+  N-Gon Hypar Origami", *Lecture Notes in Mechanical Engineering* (2026),
+  DOI 10.1007/978-981-96-8661-2_16 (extends the hypar analysis to N-gon outlines — partial progress
+  on Question 2).
+- Question 2 in general (other crease patterns, notably the circular pleat of Tabachnikov's Fig. 3)
+  remains essentially open. The 2019 paper states only that its homogenization framework "can be used
+  as a basis to investigate other corrugated origami shells, such as concentrically pleated patterns
+  with polygonal outlines". For circular rings/pleats see Mouthuy et al., *Nat. Commun.* 3:1290
+  (2012) (overcurvature-driven buckling of rings), which addresses smooth curved folds rather than
+  the PL question. No rigorous analogue of the hypar theorem for the circular pleat was found.
+
+## Work done
+
+1. Retrieved and verified the exact source statement (Arnold Math. J. site); the wave1.txt wording
+   needs no correction.
+2. Located, verified (DOI/Crossref) and read the 2019 Liu–Tachi–Paulino paper, which postdates the
+   problem list and resolves Question 1 in the asymptotic sense.
+3. Checked the key algebra myself: for
+   $\mathbf X(r,t)=(tr,(1-t)r,(2t-1)kr^2)$, one has $x^2-y^2=(t^2-(1-t)^2)r^2=(2t-1)r^2$, so indeed
+   $z=k(x^2-y^2)$; and $\xi=kr^2$ annihilates the factor $2\xi-r\xi'$ of their profile ODE.
+4. Independent local analysis of the crease pattern (my own computations):
+   - **Vertex geometry.** Take squares $Q_k$ of half-side $k$, $k=1,\dots,n$, spokes along
+     $y=\pm x$. At each interior corner vertex $(\pm k,\pm k)$ the sector angles are
+     $(135^\circ,45^\circ,45^\circ,135^\circ)$; at the center they are $4\times 90^\circ$. Both
+     satisfy Kawasaki's flat-foldability condition ($135-45+45-135=0$), so every vertex is locally
+     flat-foldable — the obstruction proved by Demaine et al. is genuinely global.
+   - **DOF count after triangulation.** Adding one diagonal per trapezoid (and the central vertex)
+     gives a triangulated disc with $V=4n+1$ vertices ($V_b=4$ boundary corners,
+     $V_i=4n-3$ interior), $E=12n-4$ edges ($E_{\mathrm{int}}=12n-8$), $F=8n-4$ triangles
+     (Euler check: $V-E+F=1$ ✓). The standard rigid-origami count
+     $\mathrm{DOF}=E_{\mathrm{int}}-3V_i=(12n-8)-3(4n-3)=1$.
+     So for **every** choice of diagonals the generic folded hypar is a 1-DOF mechanism: the folded
+     PL surface is not unique but moves along a one-parameter family — matching Liu–Tachi–Paulino's
+     parameter $k$ and their single-soft-mode eigenvalue computation. (This is the usual
+     Maxwell-type generic count, valid when the constraint Jacobian has full rank; I did not prove
+     full-rankness for this pattern.)
+   - Hence the 2^{4(n-1)} diagonal patterns each yield (generically) a 1-parameter family of PL
+     foldings; the experiments of Liu–Tachi–Paulino show physical paper selects the alternating
+     asymmetric pattern, and the homogenized limit of any of these families is the same surface
+     $z=k(x^2-y^2)$.
+
+## Result
+
+Question 1 is resolved in the literature (post-2015): with one added diagonal fold per trapezoid the
+pattern becomes a generically 1-DOF triangulated origami, and its folded shape — at any folding
+depth and independently of the diagonal pattern (among the symmetric schemes) and of the spacing of
+the squares — converges, as the pleats refine, to the genuine hyperbolic paraboloid
+$z=k(x^2-y^2)$, with $k$ a monotone function of the folding angle (Liu–Tachi–Paulino, *Nat.
+Commun.* 10:4238, 2019). Bistability with two mirror-symmetric stable states is proved in the same
+work. Question 2 (other fold-line patterns, e.g. circular pleats) is only partially addressed
+(N-gon hypar extensions; framework remarks) and remains open.
+
+## What remains
+
+- A fully discrete, rigorous description of the finite-$n$ PL folded surface for each of the
+  $2^{4(n-1)}$ diagonal patterns (existence/uniqueness of the 1-DOF motion, self-intersection,
+  which branch paper selects); currently only simulation and experiment.
+- The circular pleated surface (Tabachnikov's Fig. 3): no theorem analogous to the hypar limit is
+  known; even a conjectural limit shape seems absent from the literature.
+- General theory: for which concentric-polygon pleat patterns does the homogenized limit exist and
+  which Monge–Ampère-type PDE/ODE governs the limit shape (the hypar ODE came from an integrability
+  identity $\cos\psi'=\cos\gamma-1$ specific to right-angled symmetry planes).

research/AMR-005-0010.md

@@ -0,0 +1,50 @@
+---
+id: AMR-005-0010
+classification: PARTIAL-PROGRESS
+wording_corrected: yes
+---
+
+# AMR-005-0010 — Unbounded unicycle tracks (Finn's construction)
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Section 9 "The Unicycle Problem", Conjecture 3). Original wording, verified against the published article:
+
+> The following construction is due to D. Finn. Let $\gamma(t), t\in[0,L]$ be an arc length parameterized smooth curve in the plane which coincides with all derivatives, for $t=0$ and $t=L$, with the $x$-axis at points $(0,0)$ and $(1,0)$, respectively. One uses $\gamma$ as a "seed" trajectory of the rear wheel of a bicycle. Then the new curve $\Gamma = T(\gamma) = \gamma + \gamma'$ is also tangent to the horizontal axis with all derivatives at its end points $(1,0)$ and $(2,0)$. One can iterate this procedure yielding a smooth infinite forward bicycle trajectory $\mathcal{T}$ such that the tracks of the rear and the front wheels coincide.
+> **Conjecture 3.** Unless $\gamma$ is a straight segment, the amplitude of the curve $\mathcal{T}$ is unbounded, i.e., $\mathcal{T}$ is not contained in any horizontal strip; $\mathcal{T}$ is not a graph [sic: "grap;" in the published text]; and $\mathcal{T}$ is not embedded, that is, it starts to intersect itself.
+
+Corrections to the dataset transcription: essentially faithful; the dataset silently fixed the published typo "not a grap;" to "not a graph". Note that the conjecture has **three distinct clauses**: (i) *vertical* amplitude unbounded (not contained in any horizontal strip); (ii) $\mathcal{T}$ is eventually not a graph of a function $y=f(x)$; (iii) $\mathcal{T}$ is not embedded (self-intersections appear). Their literature status differs (see below), which the single-sentence transcription obscures.
+
+## Status / Literature
+
+All references verified via Crossref, the arXiv API, and publisher pages (abstracts/full text seen verbatim).
+
+- **Origin of the construction.** D. L. Finn, "Can a Bicycle Create a Unicycle Track?", *College Math. J.* 33 (2002), 283–292, DOI 10.1080/07468342.2002.11921954 (verified via Crossref). Finn's seed-and-iterate construction of "unicycle tracks".
+- **Oscillation growth (proved 2009).** M. Levi, S. Tabachnikov, "On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks", *Exp. Math.* 18 (2009), 173–186, DOI 10.1080/10586458.2009.10128894, arXiv:0801.4396 (DOI seen in the Crossref reference list of the source article; abstract seen via arXiv API). Establishes: each next arc $\gamma_n$ of $\mathcal{T}$ has strictly more intersections with the $x$-axis, more local extrema of the height function, and more inflection points than the previous one; also that a unicycle track cannot be extended backward indefinitely.
+- **"Not a graph" clause — PROVED (2025 preprint).** I. Molodyk, "On the Complexity of Horizontal Unitracks", arXiv:2510.10388 (v1, 12 Oct 2025; abstract and full HTML text read verbatim). Theorem 4.2: unless $\gamma_0$ is a straight segment, the iterates $\gamma_n = \varphi^n(\gamma_0)$ cannot all remain graphs of smooth functions. Theorem 4.3: the *horizontal* amplitude $H(\gamma_n)$ is non-decreasing and grows linearly, $n - c_1 \le H(\gamma_n) \le 2n - c_2$ for constants $c_1,c_2$ depending on $\gamma_0$; consequently the length of $\gamma_n$ tends to infinity. Proof idea for Theorem 4.2: assuming all $\gamma_n$ are graphs, the horizontal coordinates $x_n(t)$ form a decreasing sequence with a monotone limit $L$; slope estimates via $s_n(x) = 1 - \cos\arctan f_n'(x)$ and the length-monotonicity $\operatorname{Len}(\varphi(\gamma)) \ge \operatorname{Len}(\gamma)$ (strict unless $\gamma$ is straight) force $\operatorname{Len}(\gamma_0) \le 1$, which for a curve joining $(0,0)$ to $(1,0)$ forces $\gamma_0$ to be the straight segment. The author acknowledges Tabachnikov as advisor. **Caveat: preprint, not yet peer-reviewed as of this writing.**
+- **Vertical amplitude and self-intersection clauses — still OPEN.** In the same preprint these are stated explicitly as open: Conjecture 4.4 ($V(\gamma_n)$ unbounded — exactly clause (i) of Tabachnikov's Conjecture 3), Conjecture 4.5 (some $\gamma_n$ has self-intersections) and the weaker Conjecture 4.6 (the full track $\mathcal{T}$ self-intersects — clause (iii)). Molodyk does prove $V(\gamma_n)$ is (strictly) increasing for non-trivial seeds.
+- **Related recent work.** S. Wagon, "A Spiral Bicycle Track that Can Be Traced by a Unicycle", arXiv:2503.11847 (2025; abstract seen via arXiv API): numerical evidence (unibike error $<10^{-7}$) that iterating Finn's construction on the polar square-root curve converges to a spiral-shaped unibike curve — consistent with, but not resolving, the growth conjectures.
+- **Discrete analogues.** A SUMMER@ICERM 2012 undergraduate report, "On Bicycle Uni-track Path Efficiency: Bicycle 'Unicycle' Paths" (icerm.brown.edu/summerug/2012/cmj_bicycle_unicycle_paths.pdf), claims proofs of amplitude growth and failure of embedding for *discrete* unicycle paths built from line segments and circle arcs. I could not extract the PDF (fetch failed twice) and could not verify authorship or details; treat as unverified supporting evidence.
+
+## Work done
+
+- Retrieved the original statement from the published AMJ article (publisher HTML at amj.math.stonybrook.edu) and confirmed the dataset transcription is faithful (modulo the "grap;" typo fix). Verified the source article's metadata via Crossref (DOI 10.1007/s40598-014-0001-3).
+- Verified Finn (2002) and Levi–Tabachnikov (2009) via Crossref; verified arXiv records 0801.4396, 2503.11847, 1602.06455, 1211.2345 via the arXiv API.
+- Read the full text of arXiv:2510.10388 (Molodyk, Oct 2025), which is the decisive recent progress, and mapped its Theorems 4.2/4.3 and Conjectures 4.4–4.6 onto the three clauses of Tabachnikov's Conjecture 3.
+- Reasoned about the remaining open clause (i): the signed area between $\gamma_n$ and the $x$-axis is preserved under iteration (Theorem C of Molodyk's introduction), while the number of zeros of $\gamma_n$ strictly increases (Levi–Tabachnikov). These two facts are *consistent* with bounded vertical amplitude (ever denser oscillations of bounded height), so the area invariant alone cannot force $V(\gamma_n)\to\infty$; any proof must exploit finer structure (e.g. curvature blow-up near the vertical tangencies whose existence Molodyk proves).
+
+## Result
+
+The conjecture is **partially resolved in the literature** as of October 2025:
+
+1. **Clause (ii) "not a graph" — settled affirmatively** (Molodyk, arXiv:2510.10388, Theorem 4.2): every non-trivial seed eventually produces an iterate that is not a graph of a function.
+2. **A strong quantitative substitute for unboundedness** (ibid., Theorem 4.3): the horizontal amplitude grows linearly, $n-c_1 \le H(\gamma_n) \le 2n-c_2$, so in the horizontal direction the track escapes every vertical strip; lengths of the arcs tend to infinity.
+3. **Clause (i) "not contained in any horizontal strip" — open**: $V(\gamma_n)$ is known to be strictly increasing, but unboundedness is Conjecture 4.4 of the 2025 preprint.
+4. **Clause (iii) "not embedded" — open**: even the weaker statement that the full track $\mathcal{T}$ self-intersects is listed as open (Conjectures 4.5–4.6 ibid.). Levi–Tabachnikov's growth of zeros, extrema and inflection points, plus numerical evidence (Wagon 2025), strongly support it.
+
+## What remains
+
+- Prove $V(\gamma_n) \to \infty$ (vertical amplitude), the literal "amplitude" clause of Conjecture 3. Obstacle identified above: the preserved signed area and the growing oscillation count do not by themselves preclude bounded height.
+- Prove self-intersection of $\mathcal{T}$ (Molodyk's Conjectures 4.6, and the stronger 4.5). Molodyk's vertical-tangency mechanism (the leftmost point of $\gamma_n$ has vertical tangent for all large $n$ and marches left by 1–2 units per step) looks like the natural entry point: the track folds back over earlier arcs, but a rigorous intersection argument is missing.
+- Peer review / publication status of arXiv:2510.10388 should be checked before citing clause (ii) as a theorem in the strongest sense.
+- The analogous conjecture for the *circular* version (Conjecture 4 of the source: iterates of $\gamma \mapsto$ endpoints of unit tangent segments all convex $\Rightarrow$ circle) is a separate open item, not treated here.

research/AMR-005-0011.md

@@ -0,0 +1,157 @@
+---
+id: AMR-005-0011
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+# AMR-005-0011 — Convex tangent-segment iteration (bicycle front-track map)
+
+## Problem (corrected statement if needed)
+
+The transcription was checked against the source (S. Tabachnikov, "A Baker's Dozen
+of Problems", Arnold Math. J. 1 (2015), §9, Conjecture 4) and is accurate; no
+correction needed. Original statement:
+
+> Given an oriented oval $\gamma$, draw the unit tangent segments to $\gamma$, and
+> let $\gamma_1$ be the locus of their endpoints. We get a map
+> $\gamma\mapsto\gamma_1$. **Conjecture:** If all iterations of this map are convex
+> curves then $\gamma$ is a circle.
+
+Equivalently: parametrize the oval by its tangent angle $\theta$,
+$X'(\theta)=\rho(\theta)(\cos\theta,\sin\theta)$ with radius of curvature
+$\rho>0$ and closure condition $\int_0^{2\pi}\rho\,e^{i\theta}\,d\theta=0$.
+The map is $Y(\theta)=X(\theta)+(\cos\theta,\sin\theta)$ (in bicycle language:
+$\gamma$ is the rear-wheel track, $\gamma_1$ the front-wheel track of a unit
+segment). Conjecture: the only ovals whose whole forward orbit stays convex are
+circles. Circles do survive: $\rho\equiv R\Rightarrow \rho_1\equiv\sqrt{1+R^2}$.
+
+## Status / Literature
+
+- Source: S. Tabachnikov, *A Baker's Dozen of Problems*, Arnold Math. J. 1 (2015),
+  §9, Conjecture 4 (verified at the journal page:
+  https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/).
+  Tabachnikov's justification: the linearized statement is a theorem — if $F$ is
+  periodic and all iterates of $F\mapsto F+F'$ are positive, then $F$ is a
+  positive constant.
+- Related literature on the same map (bicycle/unicycle kinematics):
+  M. Levi & S. Tabachnikov, *On bicycle tire tracks geometry, hatchet planimeter,
+  Menzin's conjecture and oscillation of unicycle tracks*, Exp. Math. 18 (2009)
+  173–186 — for the open-arc ("unicycle") version they prove the number of local
+  extrema and of inflection points strictly increases at each iteration, i.e.
+  complexity grows under the same map;
+  R. Foote, M. Levi, S. Tabachnikov, *Tractrices, bicycle tire tracks, hatchet
+  planimeters, and a 100-year-old conjecture*, Amer. Math. Monthly 120 (2013)
+  199–216;
+  G. Bor, M. Levi, R. Perline, S. Tabachnikov, *Tire tracks and integrable curve
+  evolution*, arXiv:1705.06314 (IMRN 2021) — relates the continuous bicycle flow
+  to the AKNS system and the filament equation (checked the abstract; it does not
+  address this convexity conjecture).
+- I found **no published resolution** of Conjecture 4. Caveat: web search was
+  heavily rate-limited during this work (repeated HTTP 429 from the search
+  provider and arXiv/Semantic Scholar APIs); the negative finding is based on
+  the source article, the arXiv abstract of arXiv:1705.06314, and one successful
+  DuckDuckGo query, so a very recent resolution cannot be fully excluded.
+
+## Work done
+
+All derivations below are mine and were checked numerically.
+
+**1. Exact convexity criterion and curvature evolution.** With the tangent-angle
+parametrization,
+$$Y'(\theta)=(\rho\cos\theta-\sin\theta,\ \rho\sin\theta+\cos\theta),\quad
+|Y'|=\sqrt{1+\rho^2}>0,$$
+so the tangent angle of $\gamma_1$ is
+$\varphi(\theta)=\theta+\arctan(1/\rho(\theta))$, and its radius of curvature is
+$$\rho_1(\varphi)=\frac{(1+\rho^2)^{3/2}}{\,1+\rho^2-\rho'\,},
+\qquad \varphi'=\frac{1+\rho^2-\rho'}{1+\rho^2}.$$
+Since a regular closed curve with strictly monotone tangent angle of total
+increment $2\pi$ is convex, we get the sharp criterion
+$$\gamma_1\ \text{convex}\ \iff\ 1+\rho(\theta)^2-\rho'(\theta)>0\ \ \forall\theta.$$
+The conjecture is thus equivalent to: *the only positive periodic $\rho$ with
+$\int\rho e^{i\theta}=0$ whose whole forward orbit under
+$\rho\mapsto(1+\rho^2)^{3/2}/(1+\rho^2-\rho')$ (with the reparametrization
+$\varphi=\theta+\arctan(1/\rho)$) keeps satisfying $1+\rho_n^2-\rho_n'>0$ is
+$\rho\equiv\mathrm{const}$.*
+
+**2. Rigorous proof of Tabachnikov's linearized model.** Claim: if $F$ is a
+smooth $2\pi$-periodic real function and $T^nF>0$ for all $n\ge0$, where
+$TF=F+F'$, then $F$ is a positive constant. Proof: $\widehat{T^nF}(k)=(1+ik)^n\hat
+F(k)$ and the mean of $T^nF$ is $\hat F(0)$ for all $n$. If $\hat F(m)\neq0$ for
+some $m\neq0$, then
+$\sup(T^nF)-\inf(T^nF)\ge 2(1+m^2)^{n/2}|\hat F(m)|\to\infty$ while the mean
+stays $\hat F(0)$, so $T^nF$ takes negative values for large $n$ — contradiction.
+Hence $F\equiv\hat F(0)>0$. $\blacksquare$
+
+**3. Linearized instability of the circle (rigorous).** Writing
+$\rho=R+\varepsilon u(\theta)$ and linearizing the map of part 1:
+$$\rho_1(\varphi)=\sqrt{1+R^2}+\frac{\varepsilon}{\sqrt{1+R^2}}
+\big(Ru(\theta)+u'(\theta)\big)+O(\varepsilon^2),\qquad
+\varphi=\theta+c-\tfrac{\varepsilon u}{1+R^2}+O(\varepsilon^2),\ c=\arctan(1/R).$$
+The linearized operator $L_Ru=\big(Ru(\cdot-c)+u'(\cdot-c)\big)/\sqrt{1+R^2}$ is
+diagonalized by $e^{ik\theta}$ with eigenvalues
+$$\lambda_k(R)=\frac{(R+ik)e^{-ikc}}{\sqrt{1+R^2}},\qquad
+|\lambda_k|=\sqrt{\frac{R^2+k^2}{R^2+1}}.$$
+$k=\pm1$ (the closure/translation modes) are neutral, $k=0$ is damped, and every
+mode $|k|\ge2$ is **amplified**: $|\lambda_k|>1$. Since $R_n=\sqrt{R_0^2+n}$ for
+the circle orbit, the cumulative amplification of mode $k$ over $n$ steps is
+$\prod_j\sqrt{(R_j^2+k^2)/(R_j^2+1)}\asymp n^{(k^2-1)/2}$, while convexity
+survival needs $\rho_n'<1+\rho_n^2\sim n$; the linear estimate predicts breakup
+around $n\sim\varepsilon^{-2}$ for mode 2 (nonlinear breakup is observed much
+earlier). So circles are linearly unstable fixed points of the rescaled dynamics
+— the conjecture is the statement that this instability is never tamed
+nonlinearly.
+
+**4. Numerical experiment** (spectral code, `/tmp/amr_iter.py`,
+`/tmp/amr_iter4.py`; numpy/scipy in an isolated venv; grids $N=8192$–$131072$,
+spectral derivative, periodic cubic-spline resampling onto the uniform
+$\varphi$-grid, closure projection, low-pass filter):
+
+- Circle $\rho\equiv1$: iterates match $R_n=\sqrt{n+1}$ and
+  $\min(1+\rho^2-\rho')=1+R_n^2$ to machine precision over 30 steps (validates
+  the code).
+- Early growth of a mode-2 perturbation matches the linear prediction
+  $|\lambda_2(R)|=\sqrt{(R^2+4)/(R^2+1)}$ step by step (e.g. absolute amplitude
+  $\times1.581$ at $R=1$, $\times1.414$ at $R=\sqrt2$).
+- Every tested non-circular oval lost convexity within a few steps:
+  $\rho=1+0.2\cos2\theta$: lost at $n=4$; $1+0.05\cos2\theta$ and
+  $1+0.01\cos2\theta$: lost at $n=6$; $1+0.2\cos3\theta$: lost at $n=2$;
+  $1+0.05\cos4\theta$: lost at $n=1$; $2+0.3\cos2\theta$: lost at $n=3$.
+  The loss is abrupt: $\min(1+\rho^2-\rho')$ jumps from $\approx4$ to
+  $\approx-10^2$ in one step — the denominator $1+\rho^2-\rho'$ dips, $\rho_1$
+  spikes locally, and the next derivative explodes (a nonlinear sharpening
+  cascade, consistent with part 3's amplification of high modes).
+- Honest caveat: for very small amplitudes ($\le10^{-3}$) the *step* of breakdown
+  shifts with grid resolution (loss at $n=5$–$6$ depending on $N$), because
+  resampling noise in high Fourier modes is amplified $\sim k$ per step by the
+  map; the quoted growth rates and the moderate-amplitude phenomenology are
+  resolution-stable, but the exact lifetime of tiny perturbations is not
+  numerically trustworthy.
+
+## Result
+
+No full solution. Rigorous partial progress: (i) exact reformulation as a
+1-dimensional curvature dynamical system with the sharp convexity criterion
+$1+\rho^2-\rho'>0$; (ii) complete proof of the linearized model $F\mapsto F+F'$;
+(iii) rigorous linear analysis showing circles are isolated, linearly unstable
+fixed points of the rescaled dynamics, with all non-trivial Fourier modes
+($|k|\ge2$) amplified by $\sqrt{(R^2+k^2)/(R^2+1)}$ per step; (iv) numerical
+evidence that generic non-circular ovals lose convexity within a handful of
+iterations via a sharpening cascade. Combined with Levi–Tabachnikov's theorem
+that complexity (numbers of extrema/inflections) strictly increases under the
+same map for open arcs, the conjecture is very plausible but, to my knowledge,
+still open.
+
+## What remains
+
+- A proof (or counterexample) of the conjecture. Natural routes: (a) find a
+  monotone quantity (e.g. a weighted $L^2$ norm of the nonconstant Fourier part
+  of $\rho$, or a geometric functional like isoperimetric defect normalized by
+  scale) that strictly increases unless $\rho$ is constant; (b) promote the
+  linear analysis to a nonlinear instability statement (invariant-manifold
+  argument around the circle orbit $R_n=\sqrt{R_0^2+n}$); (c) use the
+  integrability machinery of Bor–Levi–Perline–Tabachnikov (the map is one step
+  of the discrete bicycle flow, tied to the discrete mKdV/AKNS hierarchy).
+- A definitive numerical lifetime law $n_*(\varepsilon,k)$ for small
+  perturbations requires noise-free high-precision numerics (spectral regridding
+  without interpolation, or extended precision), which I did not complete.
+- A more thorough literature pass once search-rate limits lift (the negative
+  literature finding here is not exhaustive).

research/AMR-005-0012.md

@@ -0,0 +1,64 @@
+---
+id: AMR-005-0012
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-005-0012 — Projectively self-dual polyhedra and polygons in higher-dimensional projective spaces
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), Section 10 "Self-Dual Curves and Surfaces", Problem 5. Verified against the publisher HTML and Crossref (DOI 10.1007/s40598-014-0001-3).
+
+Original wording:
+
+> **Problem 5.** Extend the results of Fuchs and Tabachnikov [2009] to projectively self-dual polyhedra, and to projectively self-dual polygons in multi-dimensional projective spaces.
+
+The dataset transcription is a faithful paraphrase (it only drops the explicit reference "Fuchs and Tabachnikov [2009]"), so no correction is needed.
+
+Context from the source: projective duality exchanges points of $\mathbb{RP}^2$ with lines of $(\mathbb{RP}^2)^*$; a curve (resp. polygon) $\gamma$ is *projectively self-dual* if some projective transformation $\mathbb{RP}^2 \to (\mathbb{RP}^2)^*$ takes $\gamma$ to its dual $\gamma^*$. Describing projectively self-dual curves is Arnold's problem 1994-17 (V. Arnold, *Arnold's Problems*, Springer/PHASIS, 2004). In $\mathbb{RP}^n$ a non-degenerate curve has an osculating hyperplane at each point, and the family of these hyperplanes is the dual curve in $(\mathbb{RP}^n)^*$; affine analogs replace projective duality by polar duality of star-shaped hypersurfaces.
+
+## Status / Literature
+
+All citations below were verified against Crossref and/or the arXiv API.
+
+1. **S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Math. J. 1 (2015), 59–67. DOI: 10.1007/s40598-014-0001-3.** The source list. Problem 5 (Section 10) is stated as open; the article presents it as an extension problem, not a conjecture with an expected answer.
+
+2. **D. Fuchs, S. Tabachnikov, "Self-dual polygons and self-dual curves", Funct. Anal. Other Math. 2(2–4) (2009), 203–220. DOI: 10.1007/s11853-008-0020-5; arXiv:0707.1048.** The paper whose results the problem asks to extend. Verified via Crossref and the arXiv API; the main results were read from the arXiv text. An $n$-gon with vertices $A_1, A_3, \dots$ and sides $B_2, B_4, \dots$ is *$m$-self-dual* ($m$ odd) if a projective map sends $A_i \mapsto B_{i+m}^*$ for all $i$. Main theorem: the moduli space $\mathcal{M}_{m,n}$ of $m$-self-dual $n$-gons in $\mathbb{CP}^2$ is a single point (the regular $n$-gon) if $(m,n)=1$; has dimension $(m,n)-1$ if $m<n$, $(m,n)>1$, $n \neq 2m$; $\dim \mathcal{M}_{m,2m} = m-3$; and $\dim \mathcal{M}_{n,n} = n-3$. Key structural facts: the duality is realized by a bilinear form $F$ on $\mathbb{C}^3$, which is symmetric iff $m=n$ (so $n$-self-dual $n$-gons are the ones self-dual with respect to a polarity); every pentagon is 5-self-dual; no $n$-gon with even $n$ is $n$-self-dual; every Poncelet polygon (odd $n$) is $n$-self-dual; a convex $n$-self-dual $n$-gon forces $F$ definite. For curves they do not give a classification (Arnold 1994-17 remains open in general) but construct examples: projections of constant-width-$\pi/2$ curves on $S^2$, described as Legendrian curves in the contact manifold of $S^2$; Radon curves (unit circles of normed planes with symmetric orthogonality) are projectively self-dual.
+
+3. **A. Chavez-Caliz, "Projective self-dual polygons in higher dimensions", Advances in Geometry 23(4) (2023), 567–582. DOI: 10.1515/advgeom-2023-0024; arXiv:2112.00177 (2021).** Verified via Crossref (full record) and the arXiv API. This paper directly addresses the *second half* of Problem 5: it studies the moduli space $\mathcal{M}_{m,n,k}$ of $m$-self-dual $n$-gons in $\mathbb{P}^k$, gives an explicit construction of self-dual polygons in higher dimensions, and determines $\dim \mathcal{M}_{m,n,k}$ for certain $(n,m)$. It also conjectures a higher-dimensional generalization of Clebsch's theorem (every pentagon in $\mathbb{RP}^2$ is pentagram-map invariant). The same material forms Chapter 3 of the author's PhD thesis "Topics in Projective Geometry of Polygons" (Penn State, 2022; seen in a search result at etda.libraries.psu.edu, not independently fetched).
+
+4. **The polyhedra half: no direct literature found.** Searches for projectively self-dual polyhedra/hypersurfaces turned up only restatements of the problem (Tabachnikov's ICERM 2013 undergraduate problem list, §2.5; a 2007 AIM workshop white paper "Rigidity and polyhedral combinatorics" listing "affinely and projectively self-dual polygons and polyhedra" as open — both seen as search snippets, not fetched in full). No paper extending Fuchs–Tabachnikov to polyhedra in $\mathbb{RP}^3$ appears in the citing literature of arXiv:0707.1048 (13 citing works checked via Semantic Scholar; the only directly relevant one is Chavez-Caliz).
+
+## Work done
+
+- Identified the source list and confirmed the original wording on the publisher's site (AMJ HTML), and verified the source article's bibliographic record via Crossref.
+- Verified Fuchs–Tabachnikov 2009 (Crossref DOI record + arXiv API) and read the introduction and main results from the arXiv text (Theorem 1, Propositions 2, 5, 7, 9, 13, and the curve constructions in Section 6).
+- Enumerated the citing literature of Fuchs–Tabachnikov 2009 via Semantic Scholar (13 citations) and checked each for relevance; only Chavez-Caliz 2021/2023 addresses Problem 5 (the polygon half). Verified her paper via Crossref and the arXiv API.
+- Searched specifically for projectively self-dual polyhedra/surfaces; found only restatements of the open problem.
+- Attempted original progress on the polyhedra half (below).
+
+## Result
+
+The problem splits into two halves with different statuses:
+
+**(a) Self-dual polygons in $\mathbb{RP}^k$ — partially solved in the literature.** Chavez-Caliz (2021/2023) defines $m$-self-dual $n$-gons in $\mathbb{P}^k$, constructs them explicitly, and computes $\dim \mathcal{M}_{m,n,k}$ in specific cases; the general dimension formula and her higher-dimensional Clebsch conjecture remain open.
+
+**(b) Self-dual polyhedra in $\mathbb{RP}^3$ — open; modest original progress here.** A polyhedron $P \subset \mathbb{RP}^3$ is projectively self-dual if a correlation $g: \mathbb{RP}^3 \to (\mathbb{RP}^3)^*$ takes $P$ to its dual $P^*$. Two observations, provable by hand:
+
+1. *Trivial example:* every tetrahedron is projectively self-dual (its dual is a tetrahedron, and all tetrahedra are projectively equivalent).
+
+2. *Pyramid construction (new, elementary).* For every odd $n \geq 5$, every convex $n$-self-dual $n$-gon of Fuchs–Tabachnikov gives rise to a projectively self-dual polyhedron: the pyramid over it. *Proof sketch.* Let $Q \subset H \cong \mathbb{RP}^2$ be an $n$-self-dual $n$-gon with respect to a polarity, $n$ odd, and let $a \notin H$ be the apex; write $\Pi(Q,a)$ for the pyramid. Choose coordinates so that $H = P(\langle e_1,e_2,e_3\rangle)$, $a = [e_4]$, and take the standard (Euclidean) polarity $\perp$ on $\mathbb{R}^4$. The dual polyhedron $\Pi(Q,a)^*$ has vertices dual to the faces of $\Pi(Q,a)$: the base face $H$ dualizes to $H^* = [e_4] = a$, and the side faces (planes through $a$ and the sides of $Q$) dualize to points of $a^* = H$ forming $Q^\perp$, the polar dual of $Q$ in $H$. Hence $\Pi(Q,a)^* = \Pi(Q^\perp, a)$. By Fuchs–Tabachnikov (Prop. 9 and the definite-form construction), a convex $n$-self-dual $Q$ satisfies $Q^\perp = h(Q)$ for some $h \in PO(3)$ acting on $H$; extending $h$ to $\mathbb{RP}^3$ by fixing $e_4$ gives a projective map $\tilde h$ with $\tilde h(\Pi(Q^\perp, a)) = \Pi(Q,a)$, and the correlation $g = \perp \circ \tilde h^{-1}$ realizes the self-duality. $\square$
+
+   By FT's Theorem 1, $\dim \mathcal{M}_{n,n} = n-3$, so this yields an $(n-3)$-dimensional family (plus placement freedom for the apex) of non-trivial projectively self-dual polyhedra for every odd $n \geq 5$ — the first infinite families beyond the tetrahedron, and a direct "polyhedra" analog of FT's main existence result.
+
+3. *Dimension heuristic for the general problem.* A correlation of $\mathbb{RP}^3$ is a non-degenerate bilinear form $F$ on $\mathbb{R}^4$ up to scale (15 parameters; symmetric $F$ = polarity, skew $F$ = null polarity). Self-duality of a combinatorially self-dual polyhedron with $v$ vertices, $f = v$ faces and $e$ edges imposes one bilinear incidence equation $F(v_i, v_j) = 0$ per edge (vertex $j$ lies on the polar plane of vertex $i$). With $3v$ parameters for the vertices and $\dim PGL(4) = 15$, the naive count gives a $(3v - e)$-dimensional moduli space; for the pyramid over an $n$-gon ($v = n+1$, $e = 2n$) this is $n + 3$, consistent in order of magnitude with the $n-3$ parameters of the base polygon plus the 3 parameters of the apex and the 3 of the base plane modulo $PGL(4)$. This mirrors the bilinear-form method of FT and suggests their entire Section 3–4 analysis (canonical forms of $F$, the symmetry dichotomy of their Proposition 2) has an $\mathbb{RP}^3$ analog, with skew-symmetric $F$ (null polarities, where every vertex lies in its own dual face) playing a new role with no planar counterpart.
+
+No claim is made that the pyramid construction exhausts self-dual polyhedra; combinatorially self-dual 3-polytopes are abundant (by Steinitz, self-dual planar 3-connected graphs), and the realization problem for general combinatorial types is untouched.
+
+## What remains
+
+- **Polyhedra (main open half).** Classify/describe projectively self-dual polyhedra in $\mathbb{RP}^3$: which combinatorially self-dual 3-polytopes admit projectively self-dual realizations; the analog of FT's moduli dimension theorem; the role of null polarities vs. genuine polarities; existence of a parity-type obstruction analogous to "no even $n$-gon is $n$-self-dual".
+- **Higher-dimensional polygons.** The general dimension formula for $\mathcal{M}_{m,n,k}$ beyond the cases settled by Chavez-Caliz; her conjectured higher-dimensional Clebsch theorem for the pentagram map.
+- **Smooth theory.** Arnold's problem 1994-17 itself (describe all projectively self-dual smooth curves in $\mathbb{RP}^2$) is still open — FT explicitly "do not attempt a complete classification"; even less is known for self-dual surfaces/hypersurfaces in $\mathbb{RP}^n$ and for the affine/polar-duality analogs (self-dual star-shaped hypersurfaces) mentioned at the end of the source section.
+- The question posed at the end of the FT introduction is also apparently open: can a smooth convex self-dual curve other than a conic be the oval of an algebraic curve?

research/AMR-005-0013.md

@@ -0,0 +1,117 @@
+---
+id: AMR-005-0013
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+# AMR-005-0013 — The Schwartz–Tabachnikov dodecagon configuration theorem
+
+## Problem (corrected statement if needed)
+
+The AMR statement is accurate; here is the precise mathematical content. This is
+Problem 11 ("New Configuration Theorems of Projective Geometry") of
+S. Tabachnikov, *A Baker's Dozen of Problems*, Arnold Math. J. 1 (2015), based on
+R. E. Schwartz and S. Tabachnikov, *Elementary Surprises in Projective Geometry*,
+Math. Intelligencer 32(3) (2010), arXiv:0910.1952.
+
+For an $n$-gon $P=\{p_1,\dots,p_n\}$ in $\mathbb{RP}^2$, the $k$-diagonal map
+$T_k : \mathcal{C}_n \to \mathcal{C}_n^*$ sends $P$ to the polygon in the dual plane whose
+vertices are the consecutive $k$-diagonals
+$\{\overline{p_1p_{k+1}}, \overline{p_2p_{k+2}}, \dots\}$; each $T_k$ is an involution,
+$T_1$ is projective duality, and $T_{abc} = T_a\circ T_b\circ T_c$, etc.
+
+**Statement (Figure 5 of the Baker's Dozen):** *If $P$ is a dodecagon inscribed in a
+conic, then $T_{31313}(P)$ is circumscribed about a conic; equivalently (dualizing the
+last step), $T_{131313}(P)$ is again inscribed in a conic.*
+
+This was the one case of the Schwartz–Tabachnikov configuration theorems for which the
+authors had **only numerical evidence**: the brute-force symbolic check (vertices on
+$y=x^2$, determinantal identities) was estimated at $>10^{12}$ monomials, beyond
+Mathematica. A cyclically relabeled equivalent form ($\sigma(i)=5i \bmod 12$):
+$T_{535353}(P)$ is inscribed.
+
+## Status / Literature
+
+- **Schwartz–Tabachnikov (2010), arXiv:0910.1952.** Eight configuration theorems;
+  all proved by symbolic computation *except* the starred dodecagon case
+  (Theorem 4(iii) there).
+- **Tabachnikov (2015), Baker's Dozen, §11.** Restates the dodecagon statement as
+  an open problem: "Find a proof."
+- **Tabachnikov (2016), *Projective configuration theorems: old wine into new
+  wineskins*, arXiv:1607.04758** (survey chapter; published version: in
+  *Fifty Years of Mathematics* / EMS volume, DOI 10.1007/978-3-030-13609-3_9).
+  This states explicitly (I verified the full TeX source):
+  > "Fedor Nilov proved Theorem [dodecagon case] using a planar projection of
+  > hyperboloid of one sheet. Unfortunately, none of these proofs were published."
+  So the problem was **resolved by F. Nilov**, but the proof was never written up.
+- **Izosimov (2016), *Pentagrams, inscribed polygons, and Prym varieties*,
+  ERA-MS 23 (2016), arXiv:1607.03558.** Gives a conceptual algebro-geometric proof of
+  the *related* Schwartz–Tabachnikov theorem $E_k=O_k$ for inscribed polygons
+  (self-duality $M(z)=(M(z^{-1})^{-1})^t$ of the scaled monodromy), and explicitly
+  lists "obtain an algebraic geometric explanation of [the Elementary Surprises]
+  results" as still open. I verified his full text does **not** prove the dodecagon
+  configuration theorem.
+- **Glick, *The Devron property* (2014), arXiv:1312.6881** (checked full text) and
+  **Ramassamy / Affolter Miquel-dynamics papers** (arXiv:1709.05509, 1808.04227;
+  checked full texts) do not contain the dodecagon theorem.
+- arXiv full-text searches (`"dodecagon" AND "conic"`, `"T_31313"`,
+  `"Schwartz-Tabachnikov"`) return no published proof of the dodecagon theorem.
+  Nilov's own publication list (Semantic Scholar) contains no paper on it.
+
+**Bottom line:** the conjecture is a theorem (Nilov, unpublished, reported in the
+authoritative 2016 survey by the conjecturer himself), but as far as I can determine
+no proof has ever appeared in print. Hence SOLVED-IN-LITERATURE with a caveat.
+
+## Work done
+
+1. Retrieved and read the Baker's Dozen source (AMJ site) and the underlying paper
+   arXiv:0910.1952 in full, fixing the exact statement (it is the starred case of
+   Theorem 4 there).
+2. Pulled the full TeX sources of Tabachnikov's 2016 survey, Izosimov 2016,
+   Glick 2014, Ramassamy 2018, Affolter 2018 and grepped them for the dodecagon
+   statement; cross-checked with arXiv API full-text searches and Semantic Scholar
+   citation/author queries for F. Nilov.
+3. Analysis of why the naive approaches fail / what a conceptual proof must do:
+   - The statement is a polynomial identity in the 9 cross-ratios parameterizing
+     inscribed 12-gons mod $PGL_3$; direct expansion is intractable
+     ($>10^{12}$ terms, per Schwartz–Tabachnikov).
+   - The word $w=31313$ is palindromic, so $T_{31313}$ is an involution; the space of
+     inscribed 12-gons mod projectivities is 9-dimensional, as is the space of
+     circumscribed 12-gons, so the statement is a birational "porism-type"
+     correspondence, not a dimension accident.
+   - Plausible reconstruction of Nilov's argument (**speculation, labeled as such**):
+     a one-sheeted hyperboloid $H\subset\mathbb{RP}^3$ is doubly ruled; projecting
+     $H$ from a point to a plane sends the two rulings to two families of lines
+     tangent to conics, and plane sections of $H$ to conics. A 12-gon inscribed in a
+     conic can be lifted to 12 points on $H$; the iterated diagonal intersections in
+     $T_{31313}$ lift to incidence constructions among lines of the two rulings, and
+     the final concyclicity reduces to elementary regulus geometry (the same
+     "skewers"/hyperboloid technology Tabachnikov and Nilov–Skopenkov use elsewhere).
+     This is consistent with the survey's one-line description but I did not verify
+     the details.
+   - Alternative conceptual route (open per Izosimov): the dodecagon theorem should
+     follow from algebro-geometric properties of the pentagram-map spectral curve of
+     inscribed polygons (the Prym variety), but no one has carried this out.
+
+## Result
+
+The problem (including the headline dodecagon statement) is **solved**: all eight
+Schwartz–Tabachnikov configuration theorems are theorems. The dodecagon case —
+the only one open at the time of the source list — was proved by **Fedor Nilov**
+using a planar projection of a one-sheeted hyperboloid; this is documented in
+Tabachnikov's 2016 survey (arXiv:1607.04758, §"Configurations"), which also notes
+that the proof was never published. No published proof of the dodecagon theorem
+appears to exist as of this search (checked: arXiv full-text search, Semantic
+Scholar, citing literature of both source papers, Nilov's own publications).
+
+## What remains
+
+- A *published* proof of the dodecagon theorem: either Nilov's hyperboloid argument
+  written up, or an independent one. The statement remains a perfectly good
+  target for a clean geometric or computer-algebra proof (modern Gröbner-basis /
+  resultant software might now handle the $>10^{12}$-term identity).
+- A conceptual algebro-geometric explanation (Prym varieties / integrable systems),
+  explicitly posed as open by Izosimov (2016).
+- The conjecture that the Schwartz–Tabachnikov list is *exhaustive* — no further
+  "surprises" of this form for $n>12$ — remains unproved.
+- Generalizations: which palindromic words $w$ in $\{T_k\}$ have the property that
+  $T_w$ maps inscribed $n$-gons to circumscribed ones?

research/AMR-005-0014.md

@@ -0,0 +1,52 @@
+---
+id: AMR-005-0014
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-005-0014 — A totally skew 3-disc in R^7
+
+## Problem (corrected statement if needed)
+
+Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), DOI 10.1007/s40598-014-0001-3 — Problem 6, Section 12 ("Totally Skew 3-Dimensional Disc in 7-Dimensional Space?"). Verified against the [publisher HTML](https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/) and [Crossref](https://api.crossref.org/works/10.1007/s40598-014-0001-3).
+
+Original wording:
+
+> A submanifold $M^k \subset \mathbb{R}^n$ is called totally skew if, for every two distinct points $x,y \in M$, the tangent spaces at these points are in general position (i.e., their affine span has dimension $2k+1$). Clearly, a necessary condition for being totally skew is $n \geq 2k+1$. It is proved in Ghomi and Tabachnikov [2008] that if $M^k$ is a totally skew disc in $\mathbb{R}^{2k+1}$ then $k \in \{1,3,7\}$. For $k=1$, a simple example is given by the cubic curve $(t, t^2, t^3)$, $t \in \mathbb{R}$.
+>
+> **Problem 6.** Is there a totally skew 3-disc in $\mathbb{R}^7$?
+
+The dataset transcription ("Does there exist a totally skew embedded 3-disc in $\mathbb{R}^7$?") is faithful to the original; no correction needed. Explicitly, an embedding $f: D^3 \to \mathbb{R}^7$ is totally skew iff for all $x \neq y$ the affine tangent 3-planes $T_x, T_y$ satisfy $\dim(T_x + T_y + \operatorname{span}(f(y)-f(x))) = 7$, equivalently (i) the direction subspaces intersect trivially and (ii) $f(y)-f(x) \notin T_x + T_y$.
+
+## Status / Literature
+
+**Open** (as of August 2026). Verified sources:
+
+- M. Ghomi, S. Tabachnikov, "Totally skew embeddings of manifolds", Math. Z. 258(3), 499–512 (2008). DOI 10.1007/s00209-007-0182-8 (verified via Crossref); preprint arXiv:math/0307044 (verified via arXiv API). Establishes: the basic theory of totally skew embeddings; the least ambient dimension $N(M)$ satisfies $N(M) \geq 2n+1$; generic maps $M^n \to \mathbb{R}^{4n+1}$ are totally skew; and the key restriction: a totally skew $k$-disc in $\mathbb{R}^{2k+1}$ can exist only for $k \in \{1,3,7\}$. The proof relates totally skew discs to nonsingular bilinear maps and the generalized vector field problem; the restriction $k+1 \in \{2,4,8\}$ comes from Adams's solution of the Hopf invariant one problem (a totally skew disc produces data of Hopf-construction type). This leaves $k=3$ (and $k=7$) as the undecided cases — exactly Tabachnikov's Problem 6.
+- D. Baralić, P. Đorđević, G. Stojanović, R. Živaljević, "Topological obstructions to totally skew embeddings", arXiv:1005.3709 (2010; abstract states acceptance in Trans. Amer. Math. Soc.). Establishes: obstructions to totally skew embeddings via the geometric dimension of the stable normal bundle of the configuration space $F_2(M)$; conjectures every compact $M^n$ ($n>1$) embeds totally skew in $\mathbb{R}^{4n-2\alpha(n)+1}$, $\alpha(n)$ = binary digit sum. Does not address the minimal-dimension disc problem.
+- Z. Norfolk, "A Local Condition for Totally Skew Embeddings", arXiv:2410.20467 (2024; verified via arXiv API and HTML version). Establishes: a third-order differential condition (an analogue of nonzero torsion) guaranteeing local total skewness; an explicit cubic polynomial $\mathbb{R}^n \to \mathbb{R}^{3n}$ giving totally skew small $n$-discs in $\mathbb{R}^{3n}$; determination of $N(\mathbb{R}^n)$ for $n$ a power of 2 (Corollary 3.1.2). Crucially, the paper states that before this work $N(M)$ was known only for $\mathbb{R}^1$ ($=3$), $S^1$ ($=4$), and $\mathbb{R}^2$ ($=6$) — confirming $N(\mathbb{R}^3)$, hence the totally skew 3-disc in $\mathbb{R}^7$ question, was still open in late 2024. The power-of-2 cases do not include $n=3$.
+- M. Harrison, "Introducing Totally Nonparallel Immersions", arXiv:1907.11312, published Adv. Math. 374 (2020) (verified via arXiv API). Studies the weaker notion (no parallel tangent lines); every $n$-manifold admits a totally nonparallel immersion in $\mathbb{R}^{4n-1}$; $TN(\mathbb{R}P^n) = 4n-1$ for $n$ a power of 2. The totally skew condition is strictly stronger, so this does not settle the disc problem.
+- G. Stojanović, S. Tabachnikov, "Non-existence of $n$-dimensional T-embedded discs in $\mathbb{R}^{2n}$", Comment. Math. Helv. 81(4), 877–882 (2006), DOI 10.4171/CMH/78 (verified as a Crossref-deposited reference of the Ghomi–Tabachnikov paper). A non-existence result for the closely related stronger notion of T-embedded discs in even codimension.
+
+## Work done
+
+- Fetched and read the full publisher HTML of the source list; confirmed the dataset wording matches Problem 6 verbatim.
+- Verified every citation above against Crossref or the arXiv API (Tabachnikov 2015, Ghomi–Tabachnikov 2008, Baralić et al. 2010, Norfolk 2024, Harrison 2019/2020, Stojanović–Tabachnikov 2006).
+- Searched for post-2015 work resolving the problem (web searches for "totally skew disc R^7", MathOverflow threads, citing papers of Ghomi–Tabachnikov). No solution or claim of solution found; the most recent paper in the area (Norfolk, Oct 2024) implicitly confirms the problem is open.
+- Attempted the problem directly (see Result).
+
+## Result
+
+No solution — this is a genuinely open problem. Summary of what is known and of my own analysis:
+
+1. **The obstruction side is settled.** Ghomi–Tabachnikov prove that a totally skew $k$-disc in $\mathbb{R}^{2k+1}$ exists only if $k \in \{1,3,7\}$, via a reduction to nonsingular bilinear maps and Adams's Hopf-invariant-one theorem. For $k=3$ (and $k=7$) the obstruction vanishes: nonsingular bilinear maps $\mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}^7$ do exist (quaternionic Hopf construction). So the problem sits exactly at the boundary where algebraic topology gives no answer either way.
+
+2. **My analysis of the constructive side.** Fixing a basepoint and projecting onto its normal space, a totally skew $f: D^3 \to \mathbb{R}^7$ yields a family of tangent 3-planes $\{T_x\}$ that are pairwise complementary linear subspaces; writing $T_x$ as the graph of $A_x: \mathbb{R}^3 \to \mathbb{R}^4$, one needs $A_x - A_y$ injective for all $x \neq y$ (a map of the configuration space into the Stiefel manifold $V_3(\mathbb{R}^4)$), plus the global displacement condition $f(y)-f(x) \notin T_x + T_y$. The natural first attempt, a quadratic graph $f(x) = (x, Q(x))$ with $DQ_x(v) = B(x,v)$, fails on two counts: (a) one needs a *symmetric* nonsingular bilinear $B: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^4$ for parallel-tangent freeness (the elementary candidate $B(u,v) = (u \cdot v, u \times v)$ is nonsingular but not symmetric; symmetric candidates I checked by hand, e.g. symmetrized coordinate products, all turn out singular); (b) more fundamentally, a direct computation shows the affine tangent spaces of a purely quadratic graph always intersect (one solves explicitly for the intersection parameter), consistent with the known non-existence of skew branes on nondegenerate quadrics (Sha–Solomon) and of T-embedded discs (Stojanović–Tabachnikov). Higher-order (cubic) terms are therefore essential — this is exactly the role of the torsion-like third-order condition in Norfolk's 2024 local theory, which however only produces examples in codimension $\geq 2n$ (e.g. totally skew 3-discs in $\mathbb{R}^9$), not in the critical codimension $n+1$.
+
+3. **Status of equivalent/stronger formulations.** A totally skew embedding of all of $\mathbb{R}^3$ into $\mathbb{R}^7$ would immediately give the disc by restriction; this stronger question is equally open ($N(\mathbb{R}^3) \in \{7, 8, \dots\}$ unknown; known bounds $7 \leq N(\mathbb{R}^3) \leq 9$ from the general lower bound and Norfolk's $\mathbb{R}^{3n}$ construction).
+
+## What remains
+
+- The core question: construct a totally skew 3-disc in $\mathbb{R}^7$ (equivalently a smooth $f$ with $\det[Df_x, Df_y, f(y)-f(x)] \neq 0$ for all $x \neq y$), or prove non-existence. Both directions seem to require new ideas: the known topological obstructions are exhausted (they yield only $k \in \{1,3,7\}$), and known local/perturbative constructions lose one or two dimensions.
+- Natural next steps: (i) try to exploit the quaternionic nonsingular bilinear map $\mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}^7$ as the second-order jet of a candidate embedding and control the third-order (torsion-type) term à la Norfolk in the critical codimension; (ii) investigate whether Norfolk's local condition can be satisfied by a map $\mathbb{R}^3 \to \mathbb{R}^7$ (the space of cubic polynomials modulo the discriminant is small here, so this is a concrete finite-dimensional algebraic question); (iii) the same question for $k=7$ in $\mathbb{R}^{15}$, presumably harder.
+- Related open problem: determine $N(\mathbb{R}^n)$ for $n$ not a power of 2, in particular $N(\mathbb{R}^3)$.

research/AMR-005-0015.md

@@ -0,0 +1,54 @@
+---
+id: AMR-005-0015
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-005-0015 — Polynomial relations among triangle areas in a dissection of a square
+
+## Problem (corrected statement if needed)
+
+Source: Serge Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), DOI [10.1007/s40598-014-0001-3](https://doi.org/10.1007/s40598-014-0001-3), §13 "Areas and Dissections into Triangles", **Problem 7** (verified against the publisher HTML at amj.math.stonybrook.edu and Crossref).
+
+Original wording: Consider a partition of a square (of unspecified size) into $n$ triangles. Allowing small perturbations (each interior vertex has 2 degrees of freedom, each vertex on a side has 1, plus one for scaling), the moduli space $M$ of partitions with fixed combinatorics has $\dim M = n-1$. The map $M^{n-1}\to\mathbb{R}^n$ sending a partition to the ordered tuple of triangle areas is component-wise quadratic, so there is a polynomial relation among the areas $a_1,\dots,a_n$ depending only on the combinatorics. **Problem 7: What can be said about this polynomial relation? For example, how to find the least degree of this polynomial in terms of the combinatorics of the partition?**
+
+The dataset transcription is a faithful paraphrase of the published problem; no correction needed. Motivation in the source: Monsky's theorem (a square cannot be dissected into an odd number of equal-area triangles; Stein–Szabo 1994, Monsky 1990).
+
+## Status / Literature
+
+The problem is substantially answered by a research program of Aaron Abrams and James (Jamie) Pommersheim, plus related work. All items below were verified via Crossref API records and/or the arXiv API (DOIs/arXiv ids as stated).
+
+1. **A. Abrams, J. Pommersheim, "Spaces of Polygonal Triangulations and Monsky Polynomials", Discrete Comput. Geom. 51(1), 132–160 (2014).** DOI [10.1007/s00454-013-9553-6](https://doi.org/10.1007/s00454-013-9553-6); arXiv [2506.23444](https://arxiv.org/abs/2506.23444) (2025 arXiv upload of the published article). This paper — which Tabachnikov already cites — establishes: the areas of a (generalized) triangulation $T$ of a square satisfy a **single irreducible homogeneous polynomial relation $p(T)$ depending only on the combinatorics of $T$**, called the **Monsky polynomial**; and it gives **an algorithm computing a lower bound on $\deg p(T)$**, with several examples in which the algorithm computes the degree exactly. Since the relation ideal is principal, generated by the irreducible $p(T)$, the least-degree relation is exactly $p(T)$; hence "the least degree" = $\deg p(T)$.
+
+2. **A. Abrams, J. Pommersheim, "Generalized Dissections and Monsky's Theorem", Discrete Comput. Geom. 67(3), 947–983 (2022).** DOI [10.1007/s00454-021-00354-9](https://doi.org/10.1007/s00454-021-00354-9); arXiv [2006.04286](https://arxiv.org/abs/2006.04286). Establishes: the deformation space of generalized dissections (allowing flipped-orientation triangles) is an **irreducible algebraic variety**; Monsky's original relation polynomial $f$ can be chosen deformation-invariant, and a canonical pair of choices for $f$ is identified; and the striking structural theorem
+$$p(T) \equiv (a_1+a_2+\cdots+a_n)^{d} \pmod 2$$
+for some $d$ — i.e. modulo 2 the area relation is a pure power of the total area. This recovers and re-contextualizes Monsky's equidissection theorem: equal areas $a_i = 1/n$ with $n$ odd force $p(T)=0$ while the mod-2 form and a 2-adic valuation argument give a contradiction. Thus the polynomial relation "knows" the parity obstruction that motivated Tabachnikov's question.
+
+3. **A. Abrams, J. Pommersheim, "An Illustrated Encyclopedia of Area Relations", European J. Math. 9(3), art. 49 (2023).** DOI [10.1007/s40879-023-00622-3](https://doi.org/10.1007/s40879-023-00622-3); arXiv [2105.00563](https://arxiv.org/abs/2105.00563). Establishes: for fixed $l$, the set $\mathcal{E}_l$ of integer polynomials arising as irreducible factors of specializations of $p_T$ obtained by zeroing out all but $l$ variables is **finite**; $\mathcal{E}_l$ is computed explicitly for $l\le 4$; and in **any** dissection of a square into $l$ triangles the areas satisfy some polynomial in $\mathcal{E}_l$. Method: the rational "area map" from the drawing space to area space, and restrictions on the closure of its image from approaches to the base locus.
+
+4. **A. Abrams, J. Pommersheim, "Integrality Relations for Polygonal Dissections", Pacific J. Math. 330(2), 199–206 (2024).** DOI [10.2140/pjm.2024.330.199](https://doi.org/10.2140/pjm.2024.330.199). Establishes: in a dissection of a parallelogram, the area of any one triangle is **integral over the ring generated by the other areas**, with integrality relations invariant under deformation; a corollary is that the area polynomials (Monsky polynomials) for parallelograms have **all leading coefficients equal to $\pm 1$**; an analogous trapezoid theorem gives a new proof of Monsky's equidissection theorem.
+
+5. Related: **J.-P. Labbé, G. Rote, G. M. Ziegler, "Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles", Exp. Math. 29(3), 253–275 (2020).** DOI [10.1080/10586458.2018.1459961](https://doi.org/10.1080/10586458.2018.1459961). Uses Monsky polynomials computationally (all combinatorial triangulations of small size) to derive quantitative discrepancy bounds: in an odd dissection the areas cannot all be nearly equal, with explicit area-difference bounds. Background: P. Monsky, "On dividing a square into triangles", Am. Math. Monthly 77(2), 161–164 (1970), DOI 10.2307/2317329 (seen as a deposited reference in the records above).
+
+## Work done
+
+- Read `worklist/AMR-005-0015.md`; fetched the publisher HTML of Tabachnikov's article and confirmed the item is §13, Problem 7, and that the dataset transcription is faithful.
+- Verified every citation against the Crossref REST API (`api.crossref.org/works/<DOI>`) or the arXiv API: items 1–5 above, plus the source article's own DOI. The arXiv record 2506.23444 explicitly notes it is the post-publication upload of the 2014 DCG paper.
+- Mathematical reasoning contributed (elementary checks and synthesis, no computation):
+  - **Why a unique least-degree relation exists.** The area map $\alpha: M^{n-1}\to\mathbb{A}^n$ has constructible image of dimension $\le n-1$; its Zariski closure is a hypersurface (Abrams–Pommersheim show the deformation/drawing space is irreducible, so the closure is an irreducible hypersurface, defined over $\mathbb{Q}$ since the map is). The relation ideal in $\mathbb{Q}[a_1,\dots,a_n]$ is therefore principal, generated by a unique (up to scalar) irreducible polynomial $p(T)$; the least degree of any relation equals $\deg p(T)$. This reduces Tabachnikov's question to: describe $p(T)$ and compute $\deg p(T)$ from the combinatorics — precisely the content of papers 1–4.
+  - **Homogeneity.** Scaling the square by $\lambda$ scales every triangle area by $\lambda^2$, so the image is a cone and $p(T)$ is homogeneous (consistent with paper 1's statement). With paper 4, one may normalize $p(T)$ to have integer coefficients and leading coefficients $\pm1$.
+  - **Hand-checked small cases.** (i) $n=2$, square cut by a diagonal: $p = a_1 - a_2$, degree 1. (ii) $n=4$, one interior vertex joined to the four corners: writing $a_i$ for the triangle on side $i$, each $a_i = \tfrac12 s\,d_i$ with $d_i$ the distance to that side, and opposite distances sum to $s$; hence $p = a_1 + a_3 - a_2 - a_4$, again degree 1. These match the theory: linear relations occur precisely when areas are constrained by affine "side-distance" bookkeeping; genuinely nonlinear Monsky polynomials appear for richer combinatorics (the smallest examples are catalogued in paper 3, which computes $\mathcal E_l$ for $l\le 4$).
+
+## Result
+
+The problem is solved in the literature to the extent the question is posed. The definitive statements:
+
+- The relation is given by a **single irreducible homogeneous polynomial $p(T)\in\mathbb{Z}[a_1,\dots,a_n]$** (up to scalar), depending only on the combinatorics of the dissection: the Monsky polynomial (paper 1). It is invariant under deformation of the dissection (papers 2, 4), can be normalized to be monic with leading coefficients $\pm1$ in the parallelogram case (paper 4), and satisfies $p(T)\equiv (a_1+\cdots+a_n)^d \pmod 2$ (paper 2) — which explains Monsky's odd/even equidissection theorem as a corollary of the shape of the relation.
+- **Least degree:** since the relation ideal is principal, the least degree is $\deg p(T)$. Paper 1 gives a combinatorial algorithm that computes a lower bound on $\deg p(T)$ and computes the exact degree in worked examples; paper 3 adds finiteness and explicit computation of all low-width relations ($\mathcal E_l$ for $l\le 4$), and paper 5 shows the degrees/coefficients are effectively computable for all triangulations of modest size by direct enumeration. The general qualitative answer to "what can be said" is thus complete; the specific degree question has an algorithmic (not closed-form) answer.
+
+## What remains
+
+- No known **closed combinatorial formula** for $\deg p(T)$ (or for $p(T)$ itself) valid for all combinatorial dissections; the 2014 algorithm yields a lower bound, proved exact in examples, but I did not find a published theorem that it is always exact. Producing such a formula — or proving the lower bound always equals the degree — is the natural next step.
+- $\mathcal E_l$ is computed only for $l\le 4$; extending the encyclopedia, and understanding growth/complexity of $p(T)$ as $n\to\infty$, is open.
+- Analogues for dissections of other polygons are partly covered (parallelograms, trapezoids in paper 4; the 2014 paper works with $n$-gons), but a systematic theory for general polygonal regions and higher-dimensional (simplex-volume) analogues appears largely undeveloped.
+- Caveat: my summary of each paper's content is based on its verified abstract and bibliographic record, not on a line-by-line reading of the full texts; the precise hypotheses (e.g. the class of "generalized triangulations" needed for irreducibility/uniqueness statements) should be checked in the papers themselves before being quoted in a proof.

research/AMR-010-0101.md

@@ -0,0 +1,127 @@
+---
+id: AMR-010-0101
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0101 — Bestvina Q 1.1: finite K(G,1), no Baumslag–Solitar subgroups ⟹ hyperbolic?
+
+## Problem (corrected statement if needed)
+
+From M. Bestvina, *Questions in Geometric Group Theory* (2004), Question 1.1
+(https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf):
+
+> Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any
+> Baumslag–Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic?
+> If $G$ embeds in a hyperbolic group, is it hyperbolic?
+
+The transcription in the source file is accurate; no correction was needed.
+The condition "no $BS(m,n)$" is a *necessary* condition for hyperbolicity
+(hyperbolic groups contain no $BS(m,n)$: $BS(m,\pm m)$ contains $\mathbb{Z}^2$
+up to finite index, and $BS(m,n)$ with $|m|\neq|n|$ is solvable but not
+virtually cyclic); the question asks whether it is sufficient, given a finite
+$K(G,1)$.
+
+## Status / Literature
+
+**Both questions are answered negatively** by:
+
+- G. Italiano, B. Martelli, M. Migliorini, *Hyperbolic 5-manifolds that fiber
+  over $S^1$*, Invent. Math. **231** (2023), 1–38; arXiv:2105.14795
+  (first posted May 2021). Publication venue confirmed on B. Martelli's
+  publication list (open access: doi 10.1007/s00222-022-01141-w).
+
+The paper's Corollary 2 states: *there is a hyperbolic group $G$ containing a
+subgroup $H$ of finite type that is not hyperbolic*, where "finite type" is
+defined exactly as "fundamental group of a finite aspherical cell complex",
+i.e. $H$ admits a finite $K(H,1)$ (in fact $\mathrm{cd}(H)=4$,
+$\mathrm{cd}(G)=5$). This answers the **second** question. Corollary 3 states:
+*there is a finite type group $H$ that is not hyperbolic and does not contain
+any Baumslag–Solitar subgroup $BS(m,n)$* — immediate since $H$ lies inside a
+hyperbolic group. This answers the **first** question. The authors explicitly
+note that the pair $H<G$ was raised as Bestvina's Question 1.1 (and also as
+Brady's Question 7.2, Bridson's Question 4.1, and by Jankiewicz–Norin–Wise),
+and Corollary 3 as a question of Bestvina, Bridson (Q 2.22), and
+Druțu–Kapovich (Problem 11.129).
+
+Prior milestone: N. Brady, *Branched coverings of cubical complexes and
+subgroups of hyperbolic groups*, J. London Math. Soc. **60** (1999), 461–480,
+built a finitely presented non-hyperbolic subgroup of a hyperbolic group, but
+his group is not of type $FP_3$, hence has no finite $K(G,1)$ — so the
+finite-$K(G,1)$ version remained open until 2021.
+
+Known *positive* special cases (both cited in IMM): every finite type subgroup
+of a hyperbolic group of cohomological dimension 2 is hyperbolic (Gersten);
+and within the classes of free-by-cyclic groups (Brinkmann) and ascending HNN
+extensions of free groups, "hyperbolic ⟺ contains no Baumslag–Solitar
+subgroup" holds.
+
+## Work done
+
+I verified the statement against Bestvina's list, located the resolving paper
+via arXiv search, and read the construction and proofs in the published
+argument (arXiv:2105.14795v4 full text). Sketch of the counterexample:
+
+1. **A fibering hyperbolic 5-manifold.** Using Bestvina–Brady PL Morse theory
+   and the states/moves game of Jankiewicz–Norin–Wise on the cubulation dual
+   to a tessellation by the right-angled hyperbolic 5-polytope $P^5$, they
+   produce a cusped finite-volume hyperbolic 5-manifold $M^5$ (commensurable
+   with the Ratcliffe–Tschantz manifold, the smallest known hyperbolic
+   5-manifold) that fibers over $S^1$. The fiber $F^4$ is aspherical with
+   $\chi(F)=1$.
+2. **Killing the cusps by filling.** Truncate $M^5$ to a compact $\bar M^5$
+   whose boundary consists of flat 4-tori; isotope the fibration so it
+   restricts on each boundary 4-torus to a fibration by geodesic 3-tori. Pass
+   to a finite cover so all boundary 4-tori have systole $>2\pi$, then shrink
+   each 3-torus fiber to a point. The resulting space $\hat M^5$ is an
+   aspherical pseudo-manifold, and by the Fujiwara–Manning filling theorem
+   (Thm 2.7 of their ref. [15]) it carries a locally CAT($-\kappa$) metric, so
+   $G=\pi_1(\hat M^5)$ is hyperbolic and torsion-free.
+3. **The subgroup $H$.** The fibration descends to $\hat M^5\to S^1$ whose
+   fiber $\hat F^4$ is $\bar F^4$ with each boundary 3-torus coned to a point.
+   $\hat F^4$ is aspherical (its product with $\mathbb{R}$ covers $\hat M^5$),
+   so $H=\pi_1(\hat F^4)=\ker(G\to\mathbb{Z})$ has a finite 4-dimensional
+   $K(H,1)$ — it is of finite type, and $H<G$.
+4. **$H$ is not hyperbolic.** $H^4(H)=\mathbb{Z}$ (pseudo-manifold top class)
+   and $\mathrm{Out}(H)$ is infinite: powers of the monodromy are nontrivial
+   in $\mathrm{Out}(H)$, using that centralizers of nontrivial elements in the
+   hyperbolic group $G$ are cyclic. If $H$ were hyperbolic, Rips theory
+   (Bestvina–Feighn, *Stable actions of groups on real trees*, Cor. 1.3) would
+   split $H$ over a cyclic subgroup; but a Mayer–Vietoris computation shows
+   this is impossible, because the vertex groups have infinite index in $H$,
+   hence are $\pi_1$ of noncompact aspherical 4-dimensional covers and have
+   $H^4=0$, contradicting $H^4(H)=\mathbb{Z}$.
+5. **No Baumslag–Solitar subgroups.** $H$ embeds in the hyperbolic group $G$,
+   and hyperbolic groups contain no $BS(m,n)$; hence $H$ is a finite type,
+   non-hyperbolic group with no Baumslag–Solitar subgroups.
+
+Note the role of the coning in step 2–3: the raw fiber group $\pi_1(\bar F^4)$
+*does* contain $\mathbb{Z}^4$ cusp subgroups (hence $BS(1,1)=\mathbb{Z}^2$);
+only after the $2\pi$-filling does one get a group inside a hyperbolic group.
+
+## Result
+
+Both parts of Bestvina's Question 1.1 have answer **no**:
+- There exists a group $H$ with a finite $K(H,1)$ (indeed a finite aspherical
+  4-dimensional complex) that contains no Baumslag–Solitar subgroup $BS(m,n)$
+  yet is not hyperbolic.
+- The same $H$ embeds in a hyperbolic group $G$ ($\pi_1$ of a filled,
+  CAT($-\kappa$) 5-dimensional pseudo-manifold, $\mathrm{cd}(G)=5$).
+
+Reference: Italiano–Martelli–Migliorini, Invent. Math. 231 (2023), 1–38,
+arXiv:2105.14795, Corollaries 2 and 3.
+
+## What remains
+
+The original questions are settled, but natural strengthenings remain open
+(raised in IMM §4 and elsewhere):
+- Are there *closed* (compact) hyperbolic manifolds of dimension $\ge 5$ that
+  fiber over $S^1$? All known examples in dimension 5 are cusped; a closed
+  fibering example would give a counterexample $H$ that is a closed aspherical
+  manifold group.
+- The minimal (cohomological) dimension of such a counterexample: the IMM
+  example has $\mathrm{cd}(H)=4$ inside $\mathrm{cd}(G)=5$, while finite type
+  subgroups of hyperbolic groups with $\mathrm{cd}=2$ are always hyperbolic
+  (Gersten). The intermediate dimensions and the question whether $H$ can be
+  taken to be a *manifold* group remain of interest (cf. subsequent work of
+  the same authors on algebraic fibering up to dimension 8).

research/AMR-010-0102.md

@@ -0,0 +1,58 @@
+---
+id: AMR-010-0102
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0102 — Finite K(G,1), no Z×Z, balanced ⇒ hyperbolic? (Bestvina Q 1.2)
+
+## Problem (corrected statement if needed)
+
+Source: M. Bestvina, *Questions in Geometric Group Theory* (major revision Aug 2000, updated July 2004), Question 1.2, author-hosted PDF: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf
+
+> **Q 1.2.** Suppose G admits a finite K(G,1), does not contain Z × Z, and whenever x ∈ G is an infinite order element such that x^m and x^n are conjugate, then |m| = |n|. Is G hyperbolic?
+
+The dataset transcription matches the original PDF verbatim (checked against the fetched source); **no correction needed**. The conjugacy condition is Wise's notion of a *balanced* group [Wis00, as cited in GKL below]; the two hypotheses together are what Gardam–Kielak–Logan call *weakly algebraically hyperbolic (weakly AH)*. Note that these hypotheses are implied by "G contains no Baumslag–Solitar subgroup BS(m,n)" (Bestvina's Q 1.1): BS(m,n) with |m| ≠ |n| yields conjugate powers of distinct absolute exponents, while BS(m,±m) contains Z² (for |m|=|n| one has [x, y^m] = 1; BS(1,−1) is the Klein bottle group, virtually Z²). So Q 1.2 is a weakening of Q 1.1.
+
+## Status / Literature
+
+**Resolved — the answer is NO.** All items below were verified against the arXiv API or Crossref.
+
+- **Italiano, Martelli, Migliorini**, *Hyperbolic 5-manifolds that fiber over S¹*, **Invent. Math. 231 (2023), 1–38**. DOI: [10.1007/s00222-022-01141-w](https://doi.org/10.1007/s00222-022-01141-w) (verified via Crossref); arXiv: [2105.14795](https://arxiv.org/abs/2105.14795) (verified via arXiv API). They construct finite-volume cusped hyperbolic 5-manifolds fibering over the circle (including the Ratcliffe–Tschantz manifold) and, as a consequence, "build a finite type subgroup of a hyperbolic group that is not hyperbolic" — i.e. a group G of **type F** (admitting a finite K(G,1)) embedded in a hyperbolic group, hence with no Z×Z and no Baumslag–Solitar subgroups and balanced, which is **not hyperbolic**. This is exactly a counterexample to Q 1.2 (and to Q 1.1).
+- **Gardam, Kielak, Logan**, *Algebraically hyperbolic groups*, arXiv: [2112.01331](https://arxiv.org/abs/2112.01331) (v3, 2025; "to appear in Groups, Geometry, and Dynamics" per the arXiv record; verified via arXiv API). Their introduction states explicitly: "Recently, Italiano, Martelli and Migliorini constructed a non-hyperbolic group G of type F that embeds into a hyperbolic group [IMM23, Corollary 2], which is therefore a counter-example to both of Gromov's questions. The group G they construct has geometric and cohomological dimension 4." They attribute the questions, as posed by Gromov, to Bestvina's list [Bes04, Questions 1.1 & 1.2]. They also record that the questions **remain open** for groups with a finite classifying space of dimension ≤ 3 (their Questions 1.1 and 1.2).
+- **Brady**, *Branched coverings of cubical complexes and subgroups of hyperbolic groups*, **J. London Math. Soc. (2) 60 (1999), 461–480**. DOI: [10.1112/s0024610799007644](https://doi.org/10.1112/s0024610799007644) (verified via Crossref). Gives a finitely presented non-hyperbolic subgroup of a hyperbolic group, showing the "finite K(G,1)" hypothesis cannot be relaxed to "finitely presented" (already noted in Bestvina's remarks to Q 1.1).
+- Related companion paper: Italiano–Martelli–Migliorini, *Hyperbolic manifolds that fibre algebraically up to dimension 8*, **J. Inst. Math. Jussieu 23 (2024), 609–646**, DOI: [10.1017/s1474748022000536](https://doi.org/10.1017/s1474748022000536) (verified via Crossref); arXiv:2010.10200.
+
+Positive special cases reported in the literature (as cited in the introduction of Gardam–Kielak–Logan; **not independently re-verified by me**): yes for 3-manifold groups (via Perelman's geometrization), for free-by-cyclic groups [Brinkmann 2000], for ascending HNN extensions of free groups [Mutanguha 2021], and for fundamental groups of special cube complexes [Caprace–Haglund 2009].
+
+## Work done
+
+- Read `worklist/AMR-010-0102.md`; fetched Bestvina's source PDF and confirmed the transcription of Q 1.2 is verbatim (wording_corrected: no). Note the dataset header "Source item: Question 1.2 (PDF page 2)" matches; Q 1.2 appears on PDF page 2 of the updated list.
+- Searched for the current status. Found that the question (and its strengthening Q 1.1) was resolved **negatively** by Italiano–Martelli–Migliorini; confirmed this via the abstract of arXiv:2105.14795 and, independently, via the explicit statement in Gardam–Kielak–Logan arXiv:2112.01331 (full text read), which also pins down the reference as [IMM23, Corollary 2] and the dimension of the counterexample as 4.
+- Verified all primary citations: IMM paper via Crossref (Invent. Math. 231 (2023), 1–38) and arXiv API; Brady 1999 via Crossref; Gardam–Kielak–Logan via arXiv API. (A MathOverflow thread, question 82173, on exactly Q 1.1 exists but could not be fetched — HTTP 403; status confirmed without it.)
+
+Mathematical reasoning (why "embeds in a hyperbolic group" suffices for Q 1.2's hypotheses): Let G be torsion-free and embedded in a hyperbolic group Γ.
+
+- *No Z×Z*: subgroups of hyperbolic groups contain no Z² (centralizers of infinite-order elements in Γ are virtually cyclic; a Z² would quasi-isometrically embed a Euclidean plane in a δ-hyperbolic space).
+- *Balancedness*: in a hyperbolic group every infinite-order element x has positive translation length τ(x) > 0, translation length is a conjugacy invariant, and τ(x^k) = |k|·τ(x). Hence if x^m and x^n are conjugate, |m|·τ(x) = |n|·τ(x), so |m| = |n|. The same holds in the subgroup G.
+
+Thus the IMM counterexample group — type F, non-hyperbolic, embedded in a (torsion-free) hyperbolic group — satisfies all hypotheses of Q 1.2 while failing the conclusion.
+
+## Result
+
+**The answer to Q 1.2 is NO.** Italiano–Martelli–Migliorini (Invent. Math. 231 (2023), 1–38; arXiv:2105.14795) construct a group G that:
+
+1. admits a finite K(G,1) (is of type F), in fact of geometric and cohomological dimension 4;
+2. contains no Z × Z (being a subgroup of a hyperbolic group);
+3. is balanced: x^m conjugate to x^n with x of infinite order forces |m| = |n| (translation-length argument, above);
+4. is not hyperbolic.
+
+So the class of groups satisfying Bestvina's hypotheses strictly contains the torsion-free hyperbolic groups. The same counterexample simultaneously answers Bestvina's Q 1.1 (no Baumslag–Solitar subgroups) in the negative. The construction uses circle-valued Morse theory/Bestvina–Brady-type finiteness arguments on fibering cusped hyperbolic 5-manifolds; the fiber-kernel (after suitable filling) is the non-hyperbolic type-F subgroup.
+
+## What remains
+
+- **Low-dimensional case is open**: for groups with a finite K(G,1) of dimension ≤ 3 the question is still unresolved — this is precisely Questions 1.1 and 1.2 of Gardam–Kielak–Logan (arXiv:2112.01331). The IMM counterexample has dimension 4, so the dimension bound matters.
+- **Cohomological dimension 2** (which includes Gersten's question whether every BS-free one-relator group is hyperbolic): open. For cd-2 groups, Gardam–Kielak–Logan prove weakly AH ⇔ BS-free ⇔ "algebraically hyperbolic", so the two versions coincide there.
+- **Non-positively curved 2-complex case** (Bestvina's remark after Q 1.1): if the universal cover contains a flat, must G contain Z×Z? This "flat closing" type question remains open, as does the analogous question for CAT(0) groups in general.
+- **Structural theory of the enlarged classes**: Gardam–Kielak–Logan initiate the study of (weakly) algebraically hyperbolic groups (CSA property, abelian JSJ decompositions); whether a weakly AH group of type F exists that does *not* embed in any hyperbolic group is open and tied to their Question 1.4 (must a finitely generated cyclic extension of an infinite torsion group have infinite cohomological dimension?).
+- Positive answers are known for 3-manifold groups, free-by-cyclic groups, ascending HNN extensions of free groups, and special cube complex groups (cited via Gardam–Kielak–Logan; not independently re-verified).

research/AMR-010-0105.md

@@ -0,0 +1,63 @@
+---
+id: AMR-010-0105
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0105 — Equivariant negatively curved metric on the Rips complex of a hyperbolic group (Davis)
+
+## Problem (corrected statement if needed)
+
+The dataset transcription matches the source verbatim; no correction was needed. The source is Mladen Bestvina's curated list *Questions in Geometric Group Theory* (major revision August 2000, updated July 2004), Question 1.5, hosted at the University of Utah ([questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)):
+
+> **Q 1.5. (Davis)** If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?
+
+Bestvina's list appends a remark (reproduced here in full, since the "Background" of the dataset omits it):
+
+> A potential counterexample is the mapping torus of a hyperbolic automorphism of a free group, or perhaps the quotient of a uniform lattice in $Sp(n,1)$ by a "random" element. For a related example see [...]
+
+(the trailing reference was truncated in the source extraction; from context it is presumably Gromov's *Asymptotic invariants of infinite groups* [Gro93]).
+
+Conventions: for $G$ finitely generated with word metric from a finite generating set, the Rips complex $P_d(G)$ is the flag simplicial complex whose simplices are finite subsets of $G$ of diameter $\le d$. "Equivariant negatively curved metric" means a $G$-invariant CAT($-\kappa$) metric (some $\kappa>0$), typically piecewise-hyperbolic, with the left action by isometries; for large $d$ the action is then automatically proper and cocompact. Recall Rips's theorem: for $d \ge 4\delta+2$ (with $\delta$ the hyperbolicity constant) $P_d(G)$ is contractible, and Meintrup–Schick showed it is a finite model for the universal proper $G$-space $\underline{E}G$ (D. Meintrup & T. Schick, *A model for the universal space for proper actions of a hyperbolic group*, New York J. Math. 8 (2002), 1–7; its existence and citation data were confirmed via the Crossref-deposited reference list of [Lang 2013], DOI 10.1142/S1793525313500118).
+
+## Status / Literature
+
+The question is **open**, and it is one concrete incarnation of a famous open problem, Gromov's "curvature conjecture" / *Jungentraum* ([Gro93, §7.B, p. 193], M. Gromov, *Asymptotic invariants of infinite groups*, in *Geometric Group Theory Vol. 2*, LMS Lecture Note Ser. 182, Cambridge Univ. Press, 1993): does every word-hyperbolic group act properly and cocompactly by isometries on a CAT($-1$) space? (Even the CAT(0) version is open.) Verified evidence that it remains open:
+
+- P.-E. Caprace, Y. de Cornulier, N. Monod, R. Tessera, *Amenable hyperbolic groups*, J. Eur. Math. Soc. 17 (2015), 2903–2947, DOI [10.4171/JEMS/575](https://doi.org/10.4171/jems/575) (verified via Crossref). They state: "for general hyperbolic locally compact groups (even discrete ones), it is an outstanding problem to determine if they can act properly cocompactly on any CAT(−1) (or even CAT(0)) space [Gro93, §7.B]."
+- A 2026 preprint, *The variety of group actions on all algebraic real hyperbolic spaces*, [arXiv:2603.03863](https://arxiv.org/html/2603.03863v1), states in its introduction: "Gromov's Jungentraum ([Gro93, p. 193]) is to show that every (finitely generated) hyperbolic group admits [a] geometric action ... on a CAT(−1) space. While this is wide open, it is expected to fail, but no counterexamples are known." (Recent preprint; used only as evidence of current status, not as a source of theorems.)
+- J. McCammond, *Constructing non-positively curved spaces and groups*, in *Geometric Methods in Group Theory*, Contemp. Math. 372, AMS, 2005 ([author PDF](https://web.math.ucsb.edu/~jon.mccammond/papers/cat0-survey.pdf); DOI not independently verified — my guessed DOI 10.1090/conm/372/06885 in fact resolves to a different paper, so I cite only the author copy). The survey explicitly discusses Davis's strategy: "As defined above, the Rips complex is a simplicial complex with no natural metric. One approach to the curvature conjecture would be to try and add a metric to ..." — confirming Q 1.5 is viewed as an approach to Gromov's conjecture, not a settled statement.
+
+Verified partial results in the vicinity:
+
+- S. Brown, *A gluing theorem for negatively curved complexes*, J. London Math. Soc. 93(3) (2016), 741–762, DOI [10.1112/jlms/jdw021](https://arxiv.org/abs/1510.02716) (verified via arXiv API). Consequence: hyperbolic limit groups, and hyperbolic groups whose JSJ components are fundamental groups of negatively curved 2-complexes (e.g., finite graphs of free groups with cyclic edge groups), are CAT(−1). This is progress on Gromov's conjecture for large classes, but the CAT(−1) spaces produced are glued 2-complexes, **not** the Rips complex itself.
+- N. Brady & J. Crisp, *CAT(0) and CAT(−1) dimensions of torsion free hyperbolic groups*, Comment. Math. Helv. 82(1) (2007), 61–85, DOI [10.4171/cmh/85](https://doi.org/10.4171/cmh/85) (verified via Crossref). They exhibit a free-by-cyclic group with CAT(0) dimension 2 but CAT(−1) dimension 3, and an infinite family of 2-dimensional hyperbolic groups (including a free-by-cyclic group with rank-6 free kernel) that do not act properly discontinuously by isometries on **any** proper CAT(0) space of dimension 2. This is directly relevant to Bestvina's proposed counterexample class (hyperbolic free-by-cyclic groups) and shows that CAT(−1) realizations, when they exist, may require more dimensions than the group's geometric/cohomological dimension — a warning sign for the Rips-complex version.
+- M. F. Hagen & D. T. Wise, *Cubulating hyperbolic free-by-cyclic groups: the general case*, Geom. Funct. Anal., DOI [10.1007/s00039-015-0314-y](https://arxiv.org/abs/1406.3292) (verified via arXiv API): every word-hyperbolic free-by-cyclic group $F \rtimes_\Phi \mathbb{Z}$ acts freely and cocompactly on a CAT(0) cube complex. So the candidate counterexamples are CAT(0), but this says nothing about CAT(−1): hyperbolic CAT(0) cube complexes can still fail to support any CAT(−1) structure (cf. Brady–Crisp).
+- U. Lang, *Injective hulls of certain discrete metric spaces and groups*, J. Topol. Anal. 5(3) (2013), 297–331, DOI [10.1142/S1793525313500118](https://doi.org/10.1142/S1793525313500118) (verified via Crossref): every word-hyperbolic group acts properly and cocompactly by isometries on its injective hull $E(\Gamma)$, a finite-dimensional polyhedral complex enjoying a weak (non-coarse) form of non-positive curvature — evidence "one level down" from CAT(0)/CAT(−1), and Lang explicitly relates it to this long-standing question.
+- Basic topological facts: $P_d(G)$ is contractible for $d \ge 4\delta+2$ (Rips; see Bridson–Haefliger, *Metric Spaces of Non-Positive Curvature*, Springer 1999, DOI [10.1007/978-3-662-12494-9](https://doi.org/10.1007/978-3-662-12494-9), III.Γ.3), and a finite $\underline{E}G$ (Meintrup–Schick, above).
+
+## Work done
+
+- Read `worklist/AMR-010-0105.md`; identified the source list and fetched Bestvina's PDF. The transcription is exact (Question 1.5, attributed to Davis); recovered the remark about candidate counterexamples.
+- Web-searched the status of Davis's question and of Gromov's curvature conjecture; the consistent picture across sources from 2004 to 2026 is: open, no counterexample known, no solution claimed.
+- Verified every cited item against Crossref or the arXiv API: Caprace–Cornulier–Monod–Tessera (10.4171/JEMS/575), Brown (arXiv:1510.02716 / 10.1112/jlms/jdw021), Brady–Crisp (10.4171/cmh/85), Hagen–Wise (arXiv:1406.3292 / 10.1007/s00039-015-0314-y), Lang (10.1142/S1793525313500118), Bridson–Haefliger (10.1007/978-3-662-12494-9). Two citation attempts were rejected by verification and corrected: a guessed DOI for McCammond's survey resolved to a Baumslag paper, and McCammond's survey appears not to be on arXiv under its title.
+- No computation was used; the analysis below is by hand.
+
+## Result
+
+No solution exists in the literature, and I could not solve it (a solution would resolve Gromov's conjecture). The rigorous synthesis:
+
+1. **Logical position.** A positive answer to Q 1.5 for all $G$ implies Gromov's conjecture, since $P_d(G)$ with an invariant CAT($-\kappa$) metric is a proper cocompact $G$-model (finiteness of $P_d(G)/G$ is automatic, and properness follows from Meintrup–Schick). Conversely Q 1.5 is in principle strictly stronger than Gromov's conjecture: metrics do not transfer across equivariant homotopy equivalences, so a group could be CAT(−1) on some space while its Rips complex supports no invariant CAT(−1) metric. Davis's question is thus a *canonical-model* strengthening of the curvature conjecture.
+
+2. **Why the naive approach fails (the precise obstruction).** Any $G$-invariant piecewise-hyperbolic metric on $P_d(G)$ must satisfy Gromov's link condition: every closed geodesic in the link of every simplex must have length $\ge 2\pi$. The link of a $k$-simplex $\sigma$ in $P_d(G)$ is itself a Rips-type complex — the complex of diameter-$\le d$ subsets of $G$ whose union with $\sigma$ still has diameter $\le d$ — built from the "corona" $B(x,d)\setminus N_k(\sigma)$ in the Cayley graph. As the local combinatorics of the Cayley graph grows complex (think of the thin quadrilaterals forced by long relators in quotients, or by the train-track dynamics of a free-group automorphism), these links acquire short essential loops that no uniform choice of simplex scale kills: shrinking simplices worsens angles in higher links, enlarging them breaks the homotopy type needed for contractibility. There is no known uniform combinatorial invariant of hyperbolicity that controls link girth in all dimensions simultaneously; hyperbolicity is a *coarse* condition, while the link condition is *local and dimension-dependent*. This gap is exactly why the CAT(0) analogue (equivariant CAT(0) metric on $P_d(G)$) is equally open, and why alternative canonical models (Lang's injective hull) only reach weaker curvature properties.
+
+3. **State of the candidate counterexamples.** Bestvina's proposed counterexample class — hyperbolic mapping tori $F_n \rtimes_\Phi \mathbb{Z}$ — is now known to be CAT(0) (Hagen–Wise), but Brady–Crisp show that even 2-dimensional hyperbolic (free-by-cyclic) groups can force CAT(−1) dimension 3 while being CAT(0) in dimension 2. Since for a torsion-free group of cohomological dimension $n$ the Rips complex is a model of dimension potentially much larger than $n$, a dimension-counting obstruction to Q 1.5 is not currently derivable, but the Brady–Crisp phenomenon shows the "expected" dimension is genuinely wrong in this class. The second candidate class (random quotients of uniform $Sp(n,1)$ lattices) retains property (T) from the ambient lattice; property (T) obstructs proper actions on CAT(0) cube complexes but is fully compatible with CAT(−1) actions (the $Sp(n,1)$ lattices themselves are CAT(−1) and (T)), so no known mechanism makes these counterexamples either.
+
+4. **Known positive territory.** All groups with "negatively curved 2-dimensional JSJ structure" — hyperbolic limit groups, graphs of free groups with cyclic edges — are CAT(−1) (Brown), but via ad hoc glued 2-complexes, not via $P_d(G)$. Nothing in the literature puts a negatively curved metric on the Rips complex of even a single non-elementary infinite-ended example class as far as I could verify.
+
+## What remains
+
+- The full question is open for every group not already covered by the 2-dimensional/gluing results; the first genuinely unknown cases are hyperbolic free-by-cyclic groups with fully irreducible atoroidal monodromy, and (even earlier in difficulty) whether **any** uniform $Sp(n,1)$ lattice's Rips complex carries an invariant CAT(−1) metric — the group is CAT(−1) on quaternionic hyperbolic space, but the Rips-complex metric is a separate matter.
+- A natural weakening with current traction: does $P_d(G)$ admit an invariant CAT(0) metric? This too is open and equivalent in spirit to "every hyperbolic group is CAT(0)."
+- A plausible attack on the negative side: find a hyperbolic group with a *coarse* obstruction to CAT(−1) actions (none is known; this is the bottleneck for Gromov's conjecture itself), or show the links in $P_d(G)$ of Brady–Crisp-type groups necessarily contain sub-$2\pi$ loops for all $d$ — which would refute Q 1.5 without refuting Gromov's conjecture.
+- A plausible attack on the positive side: exploit the quasi-tree / finite-complexity structure of links for specific classes (e.g., free groups, where $P_d$ is built from diameter-$d$ subsets of a tree) — even the free-group case of Q 1.5 does not appear to be written down in the literature.

research/AMR-010-0106.md

@@ -0,0 +1,65 @@
+---
+id: AMR-010-0106
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0106 — Gromov's surface subgroup question for one-ended hyperbolic groups
+
+## Problem (corrected statement if needed)
+
+Source: Mladen Bestvina, *Questions in Geometric Group Theory* (author-hosted PDF, major revision Aug 2000, updated July 2004), https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, Question 1.6 (§1.2 "Subgroups of Hyperbolic Groups"). Original wording, fetched and checked verbatim:
+
+> **Q 1.6. (Gromov)** Does every 1-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?
+
+The dataset transcription is **exact** — no correction needed. A "closed hyperbolic surface subgroup" means a subgroup isomorphic to π₁(S) for S a closed surface of genus ≥ 2. The one-ended hypothesis rules out the degenerate cases: finite groups (0 ends), virtually cyclic groups (2 ends), and nontrivial free products / splittings over finite groups (∞ ends), all of which can be word-hyperbolic without containing a closed surface group (a free group contains none, since every subgroup of a free group is free, and a closed surface group is not free; the same holds for free products of finite groups by Kurosh). Bestvina notes the question is "inspired by the well-known conjecture that closed aspherical 3-manifolds are virtually [Haken]".
+
+## Status / Literature
+
+**Open in general** as of this review (checked August 2026). The general case remains unresolved, but there is a rich body of positive partial results and a meaningful reduction. All citations below were verified against Crossref or the arXiv API during this review.
+
+- **Kahn–Markovic 2012** — *Immersing almost geodesic surfaces in a closed hyperbolic three manifold*, Ann. of Math. 175 (2012), 1127–1190. DOI: [10.4007/annals.2012.175.3.4](https://doi.org/10.4007/annals.2012.175.3.4) (verified via Crossref). Establishes the Surface Subgroup Theorem: every closed hyperbolic 3-manifold group contains a (quasiconvex, in fact immersed almost-geodesic) closed surface subgroup. This is the motivating special case of Gromov's question. (The cusped/finite-volume case is also known, by work of Masters–Zhang and of Baker–Cooper; I did not independently verify those DOIs, so I flag them as unverified here.)
+
+- **Gordon–Long–Reid 2004** — *Surface subgroups of Coxeter and Artin groups*, J. Pure Appl. Algebra 189 (2004), 135–148. DOI: [10.1016/j.jpaa.2003.10.011](https://doi.org/10.1016/j.jpaa.2003.10.011) (verified via Crossref reference record). Surface subgroups in certain hyperbolic Coxeter and Artin groups.
+
+- **Calegari 2008** — *Surface subgroups from homology*, Geom. Topol. 12 (2008), 1995–2007. DOI: [10.2140/gt.2008.12.1995](https://doi.org/10.2140/gt.2008.12.1995) (verified via Crossref reference record). Proves that graphs of free groups amalgamated over cyclic subgroups contain surface subgroups under a homological hypothesis, using stable commutator length.
+
+- **Kim–Oum 2014** — *Hyperbolic surface subgroups of one-ended doubles of free groups*, J. Topol. 7 (2014), 927–947. DOI: [10.1112/jtopol/jtu004](https://doi.org/10.1112/jtopol/jtu004) (verified via Crossref). Positive answer for one-ended doubles F *_w F when rank(F) = 2, or when the amalgamating words use every generator equally often; the abstract explicitly frames the work as attacking Gromov's question.
+
+- **Calegari–Walker 2015** — *Random groups contain surface subgroups*, J. Amer. Math. Soc. 28 (2015), 383–419. arXiv:[1304.2188](https://arxiv.org/abs/1304.2188) (verified via arXiv API; journal DOI 10.1090/S0894-0347-2014-00802-X appears in Crossref records). In Gromov's few-relators/density models, a random group — which is one-ended and hyperbolic with probability → 1 in the appropriate range — contains many quasiconvex surface subgroups. So the question has a positive answer for "generic" hyperbolic groups.
+
+- **Wilton 2018** — *Essential surfaces in graph pairs*, J. Amer. Math. Soc. 31 (2018), 893–919. DOI: [10.1090/jams/901](https://doi.org/10.1090/jams/901) (verified via Crossref, including the abstract). The strongest structural result to date: a positive answer whenever Γ is the fundamental group of a graph of free groups with cyclic edge groups, and, crucially, a **reduction** of Gromov's question: every one-ended hyperbolic group without 2-torsion contains either a quasiconvex surface subgroup or a quasiconvex **rigid** subgroup (one that does not split over a virtually cyclic subgroup). Hence, modulo the 2-torsion assumption, it suffices to resolve the question for *rigid* hyperbolic groups. The same paper also finds surface subgroups in limit groups.
+
+- **Markovic 2013** — *Criterion for Cannon's conjecture*, Geom. Funct. Anal. 23 (2013), 1035–1061. DOI: [10.1007/s00039-013-0228-5](https://doi.org/10.1007/s00039-013-0228-5) (verified via Crossref reference record). Surveys the problem (as "Problem 1.1 (Gromov)") and links it to Cannon's conjecture: a positive answer for groups with S² boundary, plus a quasiconvexity statement, would give a criterion for a hyperbolic group to be Kleinian.
+
+- **Ng 2025** — *Quasi-convex surface subgroups in some one-relator groups with torsion*, arXiv:[2510.01876](https://arxiv.org/abs/2510.01876) (verified via arXiv API; v2, June 2026). Its introduction (October 2025) describes Gromov's question as still open ("A longstanding question often attributed to Gromov asks whether every one-ended hyperbolic group contains a ... surface subgroup ... This has generated a lot of work"), confirming no full solution had appeared as of late 2025; it also cites Wilton's reduction as the current state of the art.
+
+## Work done
+
+1. Fetched the Bestvina questions PDF directly and confirmed the dataset transcription of Q 1.6 character-for-character (including the "(Gromov)" attribution and the surrounding §1.2 context).
+2. Searched the web for the current status, for claims of a full solution or counterexample (none found), and for recent activity.
+3. Verified every cited publication against Crossref (`api.crossref.org/works/...`) or the arXiv API; bibliographic details above come from those records, not from memory. Two sources I did not verify (Masters–Zhang, Baker–Cooper) are explicitly flagged as unverified.
+4. Considered whether a direct attack is feasible within this review's scope: it is not. The problem has resisted Gromov's school and two decades of geometric group theory; even the strongest known general result (Wilton's reduction) required new machinery (cycle precursors, essential surfaces in graph pairs).
+
+Mathematical reasoning about the shape of the problem (why the hypotheses are right, and where the difficulty lies):
+
+- **One-endedness is necessary and essentially sharp.** Any hyperbolic group admits a Dunwoody–Stallings splitting as a graph of groups with finite edge groups and vertex groups that are finite or one-ended; closed surface subgroups, being one-ended themselves, must lie (up to conjugacy) in one-ended vertex groups. So the question genuinely reduces to the one-ended case, and the hypothesis cannot be weakened.
+- **JSJ decomposition reduces further.** A one-ended hyperbolic group that splits over a 2-ended (virtually cyclic) subgroup has a JSJ decomposition; if all the pieces were handled (they are, when the pieces are free or surface-type), the combination problem remains — and this is exactly what Calegari, Kim–Oum, and Wilton attack. Wilton's theorem completes this line for groups without 2-torsion: either the splitting data already yields a quasiconvex surface subgroup, or the group contains a quasiconvex rigid subgroup. The residual core problem is therefore: *does every rigid one-ended hyperbolic group (without 2-torsion) contain a surface subgroup?*
+- **Why the rigid case is hard.** Rigid hyperbolic groups include fundamental groups of closed negatively curved manifolds in dimension ≥ 4 (where the Kahn–Markovic "good pants" machinery, which depends on the 2-dimensional geometry of immersed surfaces in 3-manifolds and on exponential mixing of the frame flow, does not apply), lattices in other rank-one groups (e.g. quaternionic hyperbolic lattices — many of which have property (T)-like rigidity phenomena), and Gromov–Kapovich–Kleiner-type groups with exotic boundaries (e.g. the Menger curve or Sierpiński carpet), which are frequently cited as candidate counterexamples. There is no known obstruction, but also no general construction.
+- **Consequences worth noting.** A positive answer would, combined with residual finiteness (itself a famous open problem, Q 1.15 on the same list), have structural consequences; Bridson–Conder–Reid (Israel J. Math. 214, 2016; seen in search results, not independently DOI-verified) show that if every one-ended hyperbolic group were residually finite and contained a quasiconvex surface subgroup, then certain embeddings T ≥ F with T one-ended hyperbolic and F free would be impossible.
+
+## Result
+
+**OPEN-TRIAGE.** The problem is a famous open question of Gromov, transcribed correctly from Bestvina's list, and it remains unsolved in full generality as of August 2026. The literature state is:
+
+- Solved cases: closed hyperbolic 3-manifold groups (Kahn–Markovic), graphs of free groups with cyclic edge groups (Wilton, extending Calegari and Kim–Oum for doubles), limit groups (Wilton), random groups (Calegari–Walker), various Coxeter/Artin and one-relator families (Gordon–Long–Reid; Ng 2025).
+- General reduction (Wilton 2018): without 2-torsion, the question reduces to rigid one-ended hyperbolic groups — those with no splitting over virtually cyclic subgroups.
+- No counterexample is known, and no known obstruction exists; the generic case is positive.
+
+## What remains
+
+1. **The rigid case**: prove or disprove that every rigid one-ended hyperbolic group contains a closed surface subgroup. Sub-cases of particular interest: closed negatively curved manifolds of dimension ≥ 4 (not known to contain immersed surfaces in general), Kapovich–Kleiner and related boundary-exotic hyperbolic groups (candidate counterexamples), and rigid small-cancellation groups.
+2. **Remove the 2-torsion hypothesis** in Wilton's reduction (currently a technical gap: the reduction is proved only for groups without 2-torsion).
+3. **The quasiconvex strengthening** (often called Q (A′) in the literature): does every one-ended hyperbolic group contain a *quasiconvex* surface subgroup? Even where surface subgroups are known, quasiconvexity is not always established.
+4. **Interaction with other open questions** on the same list: residual finiteness of hyperbolic groups (Q 1.15) and Cannon's conjecture (cf. Q 1.18 remarks and Markovic's criterion) — a positive answer to the surface subgroup question for S²-boundary groups would be a key input.
+5. Natural next steps for a researcher: attempt the rigid case for specific families (e.g. rigid one-relator groups, building on Ng's 2025 work; or 4-dimensional hyperbolic manifolds via new immersed-surface constructions), or seek a counterexample among rigid groups with Menger-curve boundary.

research/AMR-010-0107.md

@@ -0,0 +1,169 @@
+---
+id: AMR-010-0107
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0107 — Gromov: hyperbolic groups of dimension n with all infinite-index subgroups free
+
+## Problem (corrected statement if needed)
+
+The dataset transcription matches the source verbatim; no correction was needed.
+Original wording from M. Bestvina, *Questions in Geometric Group Theory*
+(major revision Aug 2000, updated July 2004), Question 1.7, Section 1.2
+"Subgroups of Hyperbolic Groups" (author-hosted PDF,
+<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>, fetched
+and read directly):
+
+> **Q 1.7. (Gromov)** For a given n is there an example of a hyperbolic group of
+> dimension n in which every infinite index subgroup is free? Or in which there
+> are no (quasi-convex) subgroups with codimension ≤ k for a given k ≤ n−2.
+
+Here "dimension" is understood as (virtual/rational) cohomological dimension;
+"codimension" is not defined in the list, and is plausibly meant either as
+cd(G) − cd(H) or in Sageev's sense (relative ends / limit-set codimension).
+The question is the "opposite possibility" to Gromov's Q 1.6 (does every
+1-ended hyperbolic group contain a surface subgroup?). The closely related
+Q 1.11 (Whyte) — *can a 1-ended hyperbolic group that is not virtually a
+surface group have every infinite-index subgroup free?* — is essentially the
+n = 2 sharpened form of the same question.
+
+## Status / Literature
+
+All items below were verified against Crossref or the arXiv API during this
+review.
+
+- **n = 1, 2: examples exist (classical).** Free groups (n = 1) by
+  Nielsen–Schreier. Closed hyperbolic surface groups (n = 2): every
+  infinite-index subgroup of a surface group is free (attributed to Johansson;
+  the modern homological proof is Strebel's theorem that infinite-index
+  subgroups of PD²-groups have cd ≤ 1, hence are free by Stallings–Swan):
+  R. Strebel, *A remark on subgroups of infinite index in Poincaré duality
+  groups*, Comment. Math. Helv. 52 (1977), 317–324,
+  DOI 10.1007/BF02567371 (verified via Crossref).
+
+- **Strong negative result in the cubulated case (the main recent progress).**
+  H. Wilton, *Surface groups among cubulated hyperbolic and one-relator
+  groups*, arXiv:2406.02121 (v3, Jan 2026, "final version accepted for
+  publication"; verified via arXiv API and by reading the HTML full text).
+  Theorem A: *a cubulated hyperbolic group G has a one-ended quasiconvex
+  subgroup of infinite index unless G is free or a surface group.* The author
+  states explicitly that this "answers questions of Gromov and Whyte in the
+  cubulated case [Bestvina's list, Questions 1.7 and 1.11]". Since cubulated
+  hyperbolic groups include C′(1/6) small-cancellation groups and, by the
+  Agol–Wise virtual Haken theory, all closed hyperbolic 3-manifold groups,
+  **no cubulated hyperbolic group of dimension ≥ 3 answers Q 1.7**. Theorem D
+  gives the analogous statement for one-relator groups (subgroup produced may
+  be infinitely generated). Wilton's Question 0.1 records the fully general
+  finitely-presented version as open, and his §5/§6 record that the
+  higher-dimensional (cd ≥ 3) Strebel-type picture is unresolved.
+
+- **Two-generator one-relator case.**
+  G. Gardam, D. Kielak, A. D. Logan, *The Surface Group Conjectures for
+  groups with two generators*, arXiv:2202.11093 (verified via arXiv API):
+  a two-generator one-relator group with every infinite-index subgroup free is
+  free or a surface group.
+
+- **n = 3, manifold groups ruled out.**
+  J. Kahn, V. Marković, *Immersing almost geodesic surfaces in a closed
+  hyperbolic three manifold*, Ann. of Math. 175 (2012), 1127–1190,
+  DOI 10.4007/annals.2012.175.3.4 (verified via Crossref): every closed
+  hyperbolic 3-manifold group contains a (quasiconvex) surface subgroup, so
+  closed hyperbolic 3-manifold groups never answer Q 1.7 for n = 3.
+
+- **Background for the codimension clause.**
+  M. Kapovich, B. Kleiner, *Coarse Alexander duality and duality groups*,
+  J. Differential Geom. 69 (2005), 279–352, DOI 10.4310/jdg/1121449108
+  (verified via Crossref): for a quasiconvex subgroup H of a hyperbolic group
+  G, the homology of the limit set ΛH and the topology of its complement in
+  ∂G are related by coarse Alexander duality; this is the standard tool for
+  making "codimension of a quasiconvex subgroup" precise. Codimension-1
+  quasiconvex subgroups are tied to cubulations (Sageev's construction); this
+  Sageev–Niblo–Roller theory is cited here from general knowledge, not
+  independently re-verified in this review.
+
+## Work done
+
+- Fetched and read the source PDF (Bestvina's updated questions list);
+  confirmed the dataset transcription is exact, including the trailing clause
+  "for a given k ≤ n−2".
+- Searched the web for the status of Q 1.7; identified Wilton's 2024–2026
+  paper as the decisive recent development and verified it (arXiv API record
+  plus reading the introduction of the HTML version, which explicitly cites
+  Bestvina's Questions 1.7 and 1.11 as being answered in the cubulated case).
+- Verified Gardam–Kielak–Logan (arXiv API), Strebel 1977, Kahn–Marković 2012,
+  Kapovich–Kleiner 2005 (all via Crossref/arXiv API; one initially guessed
+  DOI for Strebel was wrong — it resolved to a Kervaire–Murthy paper — and was
+  corrected via a Crossref bibliographic query).
+- Elementary deductions constraining any example G of dimension n ≥ 2 with all
+  infinite-index subgroups free (pure reasoning, no literature needed):
+  1. **G is torsion-free.** Every finite subgroup has infinite index (G is
+     infinite), hence must be free, hence trivial.
+  2. **G is freely indecomposable and 1-ended.** If G splits over a finite
+     subgroup, the vertex groups have infinite index, hence are free; a
+     graph of free groups with finite edge groups is virtually free, so
+     cd(G) ≤ 1, contradicting n ≥ 2. (Virtually-cyclic is likewise excluded.)
+  3. Consequently cd(G) equals the geometric dimension, G is a torsion-free
+     1-ended hyperbolic group, and every infinite-index subgroup has cd ≤ 1.
+     The question is thus precisely: does such a group exist in cd ≥ 3, i.e.
+     is there a "higher-dimensional Strebel phenomenon" beyond PD²-groups?
+  4. For the codimension clause with k = 1: hyperbolic groups with Kazhdan's
+     property (T) (e.g. cocompact lattices in Sp(n,1)) admit no proper
+     codimension-1 subgroups in Sageev's sense, since a codimension-1
+     subgroup yields a nontrivial action on a CAT(0) cube complex and
+     property (T) forces a fixed point (Sageev/Niblo–Roller theory;
+     cited from background knowledge, not re-verified here). So the k = 1
+     case of the second clause is essentially known, under that
+     interpretation of "codimension".
+- Combining (3) with Wilton's Theorem A and Agol's theorem (cubulated
+  hyperbolic ⟹ virtually special): **any example for n ≥ 3 must be a
+  hyperbolic group with no proper cocompact cubulation** — a class that
+  includes property-(T) hyperbolic groups and various non-cubulated
+  quotients, about whose subgroup structure very little is known.
+
+## Result
+
+The problem is **open**, with a sharp literature triage:
+
+- n = 1 (free groups) and n = 2 (closed hyperbolic surface groups) are the
+  only known examples of hyperbolic groups of dimension n with every
+  infinite-index subgroup free.
+- For n ≥ 3 the answer is negative in every class where the question is
+  understood: cubulated hyperbolic groups (Wilton, arXiv:2406.02121, which
+  covers closed hyperbolic 3-manifold groups and small-cancellation groups),
+  one-relator groups (Wilton's Theorem D; Gardam–Kielak–Logan for two
+  generators), and closed hyperbolic 3-manifold groups independently
+  (Kahn–Marković surface subgroups).
+- No construction of an n ≥ 3 example exists anywhere in the literature, and
+  Wilton explicitly records the general question (his Question 0.1, and the
+  cd ≥ 3 variants in his §6) as open. The codimension clause is likewise open
+  in general (only the k = 1 case is settled, via property (T), under the
+  Sageev interpretation).
+
+No solution or new theorem is claimed here; the contribution is the verified
+triage plus the elementary structural constraints (torsion-free, 1-ended,
+non-cubulated) on any hypothetical example.
+
+## What remains
+
+- **Core open case:** does there exist a hyperbolic group G with cd(G) = n ≥ 3
+  (equivalently dim ∂G = n − 1 ≥ 2) whose infinite-index subgroups are all
+  free? By the constraints above, any example must be torsion-free, 1-ended,
+  and admit no proper cocompact action on a CAT(0) cube complex — so the
+  question is a stress test for the reach of cubulation techniques, and a
+  negative answer in general would likely require extending Wilton's
+  Whitehead-complex/cut-width machinery beyond the cubulated world, which
+  Wilton himself describes as "well beyond current technology".
+- **Codimension clause:** for 2 ≤ k ≤ n − 2, does there exist a hyperbolic
+  group of dimension n with no quasiconvex subgroup of codimension ≤ k?
+  Nothing in the verified literature settles this; coarse Alexander duality
+  (Kapovich–Kleiner) is the natural framework, and the surface-subgroup
+  problem for higher-rank/rank-one lattices (e.g. Sp(n,1)) is a key test
+  case.
+- Natural next steps: (a) decide the question for property-(T) hyperbolic
+  groups (do cocompact lattices in Sp(n,1) or their small-cancellation
+  quotients contain infinite-index non-free — e.g. surface — subgroups?);
+  (b) extend the "strong Strebel" converse of Wilton's §5–6 to cd = 3 for
+  arbitrary (non-cubulated) hyperbolic groups; (c) check whether any
+  hyperbolic group with Menger-curve or Sierpiński boundary of dim ≥ 2 can
+  have all infinite-index subgroups free.

research/AMR-010-0108.md

@@ -0,0 +1,188 @@
+---
+id: AMR-010-0108
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0108 — Swarup's question: is a finitely presented, almost-normal, finite-height subgroup of a hyperbolic group quasiconvex?
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is faithful to the source: Question 1.8 of M. Bestvina's
+problem list *Questions in Geometric Group Theory* (major revision August 2000; the
+`questions-updated.pdf` version accessed for the dataset), attributed to G. A. Swarup:
+
+> **Q 1.8 (Swarup).** Suppose $H$ is a finitely presented subgroup of a word-hyperbolic
+> group $G$ which has finite index in its normalizer. Assume that there is $n>0$ such
+> that the intersection of $n$ distinct conjugates of $H$ is always finite. Is $H$
+> quasi-convex in $G$?
+
+The list itself adds: "The converse is a theorem of [Gitik–Mitra–Rips–Sageev]. A special
+case worth considering is when $G$ splits over $H$ when Gersten's converse of the
+combination theorem might be helpful. **Remark (Gitik):** The problem is open even when
+$H$ is malnormal in $G$." (Source PDF:
+https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf — the wording above
+was checked against the search-indexed text of the PDF; the PDF has the typo "Swarvp".)
+
+One interpretive note: "distinct conjugates" should be read as *essentially distinct*
+conjugates in the sense of Gitik–Mitra–Rips–Sageev (conjugates by elements in distinct
+cosets of $H$, or equivalently — since $[N_G(H):H]<\infty$ here — by elements in
+distinct cosets of $N_G(H)$). With this reading, the hypothesis "$n$ distinct
+conjugates always have finite intersection" says exactly that $H$ has **height**
+$\le n-1$ in $G$ in the GMRS sense. Because $[N_G(H):H]<\infty$, distinct and
+essentially distinct conjugates differ only by the bounded factor $[N_G(H):H]$, so the
+two formulations of the hypothesis are equivalent.
+
+## Status / Literature
+
+**Open** as of early 2026 — see the 2025 preprint of Halder–Sardar below, which states
+explicitly that the question remains open even for height 1 (weakly malnormal $H$).
+
+Verified sources (each checked against Crossref or the arXiv API):
+
+1. **R. Gitik, M. Mitra, E. Rips, M. Sageev, "Widths of Subgroups", Trans. Amer. Math.
+   Soc. 350(1) (1998), 321–329. DOI: 10.1090/S0002-9947-98-01792-9.** (Verified via
+   Crossref.) Introduces width/height of subgroups and proves that a **quasiconvex**
+   subgroup of a word-hyperbolic group has finite width (hence finite height). This is
+   the "converse" mentioned in Bestvina's list: quasiconvexity *implies* the
+   conjugate-intersection hypothesis of Q 1.8.
+
+2. **I. Kapovich, H. Short, "Greenberg's Theorem for Quasiconvex Subgroups of Word
+   Hyperbolic Groups", Canad. J. Math. 48(6) (1996), 1224–1244. DOI:
+   10.4153/CJM-1996-065-6.** (Verified via Crossref.) Hyperbolic-group analogue of
+   Greenberg's theorem: a quasiconvex subgroup $H$ has finite index in its
+   commensurator (virtual normalizer) $\mathrm{Comm}_G(H)$; in particular
+   $[N_G(H):H]<\infty$. Thus *all three* of Swarup's hypotheses ($H$ finitely
+   presented; $[N_G(H):H]<\infty$; finite height) are **necessary** conditions for
+   quasiconvexity; Q 1.8 asks whether they are jointly **sufficient**.
+
+3. **M. Mitra, "Height in splittings of hyperbolic groups", Proc. Indian Acad. Sci.
+   (Math. Sci.) 114(1) (2004), 39–54. DOI: 10.1007/BF02829670; arXiv:math/0403125.**
+   (Verified via Crossref and arXiv API.) Answers Swarup's question **affirmatively in
+   the split case**: if $H$ is a hyperbolic subgroup of a hyperbolic group $G$, the
+   intersection of any $n$ essentially distinct conjugates of $H$ is finite, $G$
+   splits over $H$ with hyperbolic vertex and edge groups, and the two inclusions of
+   $H$ are quasi-isometric embeddings, then $H$ is quasiconvex in $G$. The paper also
+   formulates a chain of successively stronger properties of a non-quasiconvex
+   subgroup (infinite height, strictly infinite height, a "strong" version with
+   intersections along powers of one element) and proves implications between them, so
+   that a negative answer to Swarup's question would yield subgroups with these exotic
+   intersection patterns. Caveat: the theorem presupposes $H$ hyperbolic and
+   quasi-isometrically embedded in the vertex groups — the general question, where $H$
+   is only finitely presented (and a priori possibly non-hyperbolic and distorted),
+   is untouched.
+
+4. **A. Pal, "Height in splittings of relatively hyperbolic groups", Geom. Dedicata
+   213 (2021), 121–135. DOI: 10.1007/s10711-020-00571-1.** (Verified via Crossref.)
+   Extends Mitra's split-case theorem to relatively hyperbolic groups.
+
+5. **C. Abbott, E. Martínez-Pedroza, "The quasi-isometry invariance of the Coset
+   Intersection Complex", Algebr. Geom. Topol. 26 (2026), 659–698. DOI:
+   10.2140/agt.2026.26.659; arXiv:2404.16628.** (Verified via arXiv API, including
+   journal ref.) Builds a simplicial complex encoding finite height / finite width /
+   almost malnormality and proves these properties are quasi-isometry invariants of
+   the pair $(G,H)$; the authors explicitly frame parts of their main theorem as
+   "evidence of a positive answer to Swarup's question" — i.e., they treat the
+   question as open.
+
+6. **R. Halder, P. Sardar, "Embeddings of trees of hyperbolic metric spaces and
+   Cannon–Thurston maps", arXiv:2511.12883 (v1 Nov 2025, v3 Feb 2026).** (Verified via
+   arXiv API.) States plainly: "even if $H$ is of height 1 in $G$, i.e. $H$ is weakly
+   malnormal in $G$, Swarup's question remains open. It is known only in certain
+   special cases." Proves existence of Cannon–Thurston maps for certain amalgams
+   $K_1 *_H K_2 \to G_1 *_H G_2$ (a weakening of the conclusion of quasiconvexity),
+   continuing the Mitra/Pal line.
+
+7. **M. Mitra, "Coarse extrinsic geometry: a survey", in *The Epstein Birthday
+   Schrift*, Geom. Topol. Monogr. 1 (1998), 341–364. DOI: 10.2140/gtm.1998.1.341;
+   arXiv:math/9810203.** (Verified via Crossref.) Survey that records Swarup's
+   question and the state of knowledge circa 1998.
+
+8. **I. Kapovich, "A non-quasiconvex subgroup of a hyperbolic group with an exotic
+   limit set", New York J. Math. 1 (1995).**
+   (Verified at https://nyjm.albany.edu/j/1995/1-12p.pdf.) Records the related theorem
+   attributed to Swarup: *a finitely presented one-ended subgroup of a word-hyperbolic
+   group is quasiconvex if and only if it has finite index in its virtual normalizer
+   (commensurator)*. This is the strongest known "purely algebraic" criterion for
+   quasiconvexity and is the backdrop of Q 1.8: Swarup asks whether the
+   commensurator hypothesis can be weakened to the normalizer hypothesis at the price
+   of adding finite height. (The underlying Swarup preprint appears never to have been
+   formally published; the statement survives through this citation.)
+
+## Work done
+
+- Read the dataset item; identified the source as Bestvina's problem list Q 1.8 and
+  confirmed the original wording (including Gitik's remark that the malnormal case is
+  open) against the indexed text of the author's PDF.
+- Verified every citation above against Crossref (DOIs 10.1090/S0002-9947-98-01792-9,
+  10.4153/CJM-1996-065-6, 10.1007/BF02829670, 10.1007/s10711-020-00571-1,
+  10.2140/gtm.1998.1.341) or the arXiv API (math/0403125, 2404.16628 with journal ref
+  AGT 26 (2026) 659–698, 2511.12883), plus the NYJM page for Kapovich 1995.
+- Established current status from the two most recent sources (Abbott–Martínez-Pedroza
+  2026, Halder–Sardar 2025/2026), both of which treat the question as open.
+- Analyzed the logical structure of the hypotheses (see Result).
+
+## Result
+
+The question is **open**; I cannot solve it, but the literature plus elementary
+reasoning gives a clean triage.
+
+**1. The hypotheses are exactly the known necessary conditions.** For $H$ a subgroup
+of a hyperbolic group $G$: quasiconvex $\Rightarrow$ $H$ finitely presented (standard),
+quasiconvex $\Rightarrow$ finite height/width (GMRS 1998), and quasiconvex
+$\Rightarrow$ $[\mathrm{Comm}_G(H):H]<\infty$, hence $[N_G(H):H]<\infty$
+(Kapovich–Short 1996). Swarup's question is precisely whether this conjunction of
+necessary conditions is sufficient. Degenerate cases are trivial: if $[G:H]<\infty$ or
+$H$ is finite, $H$ is quasiconvex; so the content is for infinite-index infinite $H$.
+
+**2. Reductions and equivalences.** Because $[N_G(H):H]<\infty$, the hypothesis
+"$n$ distinct conjugates have finite intersection" is the same as
+"height$(H)\le n-1$" up to the bounded factor $[N_G(H):H]$; the $n=2$ case is almost
+malnormality (malnormality in the torsion-free case). Gitik's remark in the list, and
+Halder–Sardar twenty years later, both record that **even the (weakly) malnormal case
+is open**.
+
+**3. Where the difficulty lies.** Swarup's own virtual-normalizer criterion (item 8
+above) shows that for finitely presented *one-ended* $H$, finite index in the
+*commensurator* suffices for quasiconvexity. The gap in Q 1.8 is twofold:
+(a) the normalizer can be much smaller than the commensurator — finite height is
+    meant to compensate by bounding how many conjugates can share an infinite
+    intersection (each commensurator coset yields such a conjugate), but no proof
+    currently upgrades "height $\le n-1$ + $[N_G(H):H]<\infty$" to
+    "$[\mathrm{Comm}_G(H):H]<\infty$";
+(b) $H$ is only assumed finitely presented, not one-ended or even hyperbolic, so the
+    one-ended criterion does not apply directly, and the ends/many-ended case requires
+    separate arguments (splittings of $H$ over finite groups, where Mitra's theorem is
+    exactly the relevant tool — this is presumably why the split case fell first).
+
+**4. Known distortion mechanisms do not give counterexamples.** The classical
+non-quasiconvex finitely presented subgroups of hyperbolic groups — Rips-construction
+kernels (normal of infinite index, so $[N_G(H):H]=\infty$) and Brady-type non-hyperbolic
+f.p. subgroups — violate at least one of Swarup's hypotheses in all known instances.
+Whether this is forced (positive answer) or whether more exotic constructions
+(finitely presented, distorted, yet finite-height and almost normal) exist (negative
+answer) is the unresolved core.
+
+**5. Partial positive results.** Mitra (2004) settled the split case ($G$ splits over
+$H$, inclusions quasi-isometric), Pal (2021) its relatively hyperbolic version;
+Halder–Sardar (2025) obtain Cannon–Thurston maps (a boundary-level shadow of
+quasiconvexity) for further amalgam configurations; Abbott–Martínez-Pedroza (2026)
+show the *hypotheses* of the question are robust under quasi-isometry, which they
+interpret as evidence for a positive answer.
+
+## What remains
+
+- The full question is open, and by Gitik's remark (still endorsed in the 2025
+  literature) even the case $n=2$, i.e. $H$ (almost) malnormal, finitely presented,
+  with $[N_G(H):H]<\infty$, is undecided.
+- Natural next steps: (i) settle whether "finite height + $[N_G(H):H]<\infty$" implies
+  $[\mathrm{Comm}_G(H):H]<\infty$, which by Swarup's criterion would answer the
+  one-ended case affirmatively; (ii) handle many-ended $H$ via its JSJ/grushko
+  splitting over finite groups combined with Mitra-type combination arguments;
+  (iii) on the negative side, attempt Rips/Brady-style constructions with controlled
+  height — Mitra's 2004 paper shows any counterexample must have (strictly) infinite
+  height analogues among its conjugate-intersection patterns, which constrains the
+  geometry such a construction must exhibit.
+- A confirmed answer either way would close one of the last open items of the
+  "algebraic characterization of quasiconvexity" program from the 1990s
+  (GMRS/Kapovich–Short/Swarup).

research/AMR-010-0109.md

@@ -0,0 +1,65 @@
+---
+id: AMR-010-0109
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-010-0109 — Injectivity radius going to infinity in a cover vs. quasi-isometric embedding (Mitra, Bestvina list Q 1.9)
+
+## Problem (corrected statement if needed)
+
+Source: M. Bestvina, *Questions in Geometric Group Theory* (major revision August 2000, updated July 2004), Question 1.9, p. 3 ([questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)). Original wording (verified against the author PDF):
+
+> **Q 1.9 (Mitra).** Let $X_G$ be a finite 2-complex with fundamental group $G$. Let $X_H$ be a cover corresponding to the f.p. subgroup $H$. Let $I(x)$ denote the injectivity radius of $X_H$ at $x$. Does $I(x)\to\infty$ as $x\to\infty$ imply that $H$ is quasi-isometrically embedded in $G$? A positive answer to the above question for $G$ hyperbolic would imply a positive answer to Q 1.8.
+
+The dataset transcription matches the original verbatim (including "f.p. subgroup $H$", i.e. $H$ finitely presented); **no correction needed**. Note that since $X_G$ is a *finite* 2-complex, $G=\pi_1(X_G)$ is automatically finitely presented; "f.p." in the statement refers to the subgroup $H$.
+
+The same question appears, with discussion, in M. Mitra, *Coarse extrinsic geometry: a survey*, Geom. Topol. Monogr. 1 (1998) 341–364 ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203)), where it is observed that the answer is **negative** if $G$ is allowed to be only finitely *generated* (HNN-extension example over $F(a,b,c,d)$ with a fast-growing reindexing function $f:\mathbb{N}\to\mathbb{N}$, stable letter conjugating $u_i=a^ib^i$ to $v_i=c^{f(i)}d^{f(i)}$; the free subgroup $\langle a,b\rangle$ is then distorted while the injectivity radius still escapes to infinity). So the substantive cases are: $G$ finitely presented, and especially $G$ word-hyperbolic.
+
+## Status / Literature
+
+All items below verified via the arXiv API or Crossref during this work.
+
+- **M. Mitra, *Coarse extrinsic geometry: a survey*, Geom. Topol. Monogr. 1 (1998), 341–364** ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203), verified via arXiv API: journal_ref confirmed). States the question, records the finitely-generated counterexample, and notes a positive answer for $G$ hyperbolic would answer Swarup's question (Bestvina's Q 1.8). This is the primary literature source for the problem.
+- **M. Mitra, *Height in splittings of hyperbolic groups*, Proc. Indian Acad. Sci. (Math. Sci.) 114(1) (2004), 39–54** ([arXiv:math/0403125](https://arxiv.org/abs/math/0403125), verified via arXiv API). Restates exactly this injectivity-radius question and proves the Q 1.8 conclusion ($H$ quasiconvex) under the extra hypothesis that $G$ splits over $H$ with hyperbolic vertex/edge groups and QI edge inclusions — a *partial* resolution of the motivating question Q 1.8, not of Q 1.9 itself.
+- **M. Mitra, *Cannon–Thurston maps for trees of hyperbolic metric spaces*, J. Differential Geom. 48(1) (1998), 135–164** (DOI [10.4310/jdg/1214460609](https://doi.org/10.4310/jdg/1214460609), verified via Crossref). Background: for graphs of hyperbolic groups with QI edge inclusions, vertex-group inclusions admit Cannon–Thurston (CT) maps; supplies the equivalence machinery (CT existence ⇔ uniform behavior of far-out geodesic segments) that links injectivity-radius-type hypotheses to boundary behavior.
+- **R. Gitik, M. Mitra, E. Rips, M. Sageev, *Widths of subgroups*, Trans. Amer. Math. Soc. 350 (1998)** (DOI 10.1090/S0002-9947-98-01792-9, seen as a Crossref-registered reference in the JDG paper above). This is the [GMRS98] of Bestvina's Q 1.8: finite width/height phenomena for quasiconvex subgroups — directly relevant to the reformulation in the Result section below.
+- **O. Baker, T. Riley, *Cannon–Thurston maps do not always exist*, Forum Math. Sigma 1 (2013), e3** ([arXiv:1206.0505](https://arxiv.org/abs/1206.0505), verified via arXiv API). Resolves (negatively) the related Bestvina Q 1.19: a hyperbolic subgroup of a hyperbolic group need not admit a CT map. This shows the boundary-continuation approach to distortion is subtler than hoped, but does **not** settle Q 1.9 (the injectivity-radius hypothesis is stronger/different from CT existence — e.g. fiber subgroups of fibered hyperbolic 3-manifold groups admit CT maps yet are exponentially distorted).
+- **O. Baker, T. Riley, *Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra*, Groups Geom. Dyn. 14(1) (2020), 255–282** ([arXiv:1209.0815](https://arxiv.org/abs/1209.0815), verified via arXiv API). CT maps can exist in the presence of arbitrarily heavy (primitive recursive) distortion — further evidence that CT existence and QI-embeddedness are decoupled; the injectivity-radius condition in Q 1.9 sits strictly between these notions (see Result).
+- Bestvina's list itself (July 2004 update) carries **no update/answer note on Q 1.9**, while many neighboring questions do have such notes.
+
+**Net status:** I found no publication resolving Q 1.9. The question as stated (f.p. $G$, f.p. $H$) and a fortiori the hyperbolic-$G$ case appear **open**; the finitely-generated-$G$ variant is settled negatively by Mitra's 1998 example. The motivating question Q 1.8 (Swarup) has a partial positive answer (Mitra 2004, splitting case) and is, to my knowledge, still open in general (Bestvina's list records Gitik's remark that it is open even for malnormal $H$; I did not find a later general resolution, but I did not exhaustively verify this).
+
+## Work done
+
+- Located and fetched the source list; confirmed the dataset wording is verbatim-correct and that the July 2004 update contains no status note for Q 1.9.
+- Identified the question's second appearance (with the f.g. counterexample) in Mitra's 1998 survey; verified all bibliographic data above via the arXiv API and Crossref (5 verified references; no citation is given that was not checked).
+- Searched for later resolutions (web search on the question text, on MathOverflow, on Mahan Mj's survey corpus); none found.
+- Mathematical work: unpacked the injectivity-radius hypothesis into a conjugacy/height statement, checked it against the standard distorted examples, and derived the mechanism behind Bestvina's remark that a positive answer for $G$ hyperbolic implies Q 1.8. Details below. No computation was used; all steps are elementary synthetic/coarse geometry.
+
+## Result
+
+**A reformulation (my partial progress).** Metrize $X_G$ so its 1-skeleton pull-back makes the universal cover $\widetilde X$ QI to $\mathrm{Cay}(G)$. Points of $X_H=\widetilde X/H$ are $H$-cosets, and a based loop at the coset $Hg$ of length $\ell$ is exactly an element $h\in H\setminus\{e\}$ with $|g^{-1}hg|_G=\ell$. Hence, up to bounded additive constants,
+
+$$I(Hg)=\tfrac12\min_{h\in H\setminus\{e\}}\,|g^{-1}hg|_G .$$
+
+Since only finitely many elements of $G$ have length $\le 2C$, one obtains, for **arbitrary** $G$ (no hyperbolicity needed):
+
+$$I(x)\to\infty \iff \text{for every } b\in G\setminus\{e\},\ \{\,Hg : zbz^{-1}\in H\,\} \text{ is bounded in } X_H .$$
+
+In words: **each fixed element $b\in G$ lies in only "$H$-boundedly many" conjugates $z^{-1}Hz$ of $H$.** Two immediate consequences:
+
+1. *Centralizer obstruction (necessary condition).* If $Z_G(h)$ has unbounded image in $H\backslash G$ for some $h\in H\setminus\{e\}$ (i.e. $Hz_n\to\infty$ with $z_n\in Z_G(h)$), then $I(Hz_n)\le |h|_G/2$, so $I\not\to\infty$. Hence Q 1.9 would follow from: *$H$ distorted $\Rightarrow$ some $h\in H\setminus\{e\}$ has $Z_G(h)$ unbounded mod $H$.* This is a weak "finite height" condition in the sense of Gitik–Mitra–Rips–Sageev.
+2. *Why Q 1.9 (hyperbolic case) implies Q 1.8.* Under Swarup's hypothesis — some $n$ such that any $n$ distinct conjugates of $H$ have finite intersection — every infinite-order $b$ lies in at most $n-1$ distinct conjugates $z^{-1}Hz$ (else $b$ is in an infinite... finite intersection, contradiction), so the reformulated hypothesis holds; a positive answer to Q 1.9 then gives $H$ QI-embedded, hence quasiconvex ($G$ hyperbolic), answering Q 1.8. This recovers and explains the remark in the source list.
+
+**Consistency checks against the standard distorted examples.**
+- *Fiber subgroup of a fibered hyperbolic 3-manifold group* $G=\pi_1(S)\rtimes_\varphi\mathbb Z$, $\varphi$ pseudo-Anosov, $H=\pi_1(S)$: for $g\in H$, $\varphi^n(g)=t^ngt^{-n}$, so at the coset $Ht^n$ (which escapes in $X_H$) the element $h_n=\varphi^n(g)\in H$ gives $I(Ht^n)\le\tfrac12|t^{-n}h_nt^n|_G=\tfrac12|g|_G$ — bounded. So $I\not\to\infty$, exactly as a positive answer to Q 1.9 requires; the mechanism is the centralizer obstruction (1) via $t\in Z_G$-dynamics. Note this subgroup *does* admit a CT map (Cannon–Thurston), confirming that the injectivity-radius condition is genuinely stronger than CT existence.
+- *Mitra's f.g. counterexample* shows the finite-presentedness of $G$ cannot be dropped: the escaping short loops are carried by the HNN relators $tu_it^{-1}v_i^{-1}$, which force no finite 2-complex model.
+
+I could not push the reformulation to a full proof: for hyperbolic $G$, distortion of $H$ means short $b_n\in G$ with $b_n\in H$, $|b_n|_H\to\infty$, and the question becomes whether such short $H$-elements must reappear (as a *fixed* $b$, or with centralizers) in unboundedly many conjugates of $H$. Hyperbolicity makes *individual* conjugates $zbz^{-1}$ long, so a positive answer requires controlling how the family of short distorted elements distributes across cosets — precisely the content of a uniform finite-height theorem for arbitrary (possibly distorted) f.p. subgroups, which is not in the literature I found.
+
+## What remains
+
+- The question is **open** for finitely presented $G$, and specifically for $G$ word-hyperbolic; also open: the weaker variant asking only whether $I(x)\to\infty$ implies $H$ is *hyperbolic* / admits a Cannon–Thurston map.
+- Via the reformulation, a positive answer for hyperbolic $G$ is equivalent to: *distortion of $H\le G$ forces a fixed conjugacy class (or a centralizer) to meet unboundedly many conjugates of $H$.* A promising route is to combine the annular-diagram structure of conjugacy in hyperbolic groups with the Gitik–Mitra–Rips–Sageev width theory; the obstacle is that distorted subgroups need not have finite width, and no counterexample with $I(x)\to\infty$ and distorted $H$ is known either.
+- Honest caveats: (i) I did not verify the current status of Q 1.8 (Swarup) beyond Mitra's 2004 partial answer — a full resolution of Q 1.8 would likely interact with Q 1.9; (ii) the Cannon–Thurston–Peano-curve reference (Geom. Topol. 11 (2007) 1315–1355) was seen only in reference lists, not independently Crossref-checked; (iii) absence of a resolution in the literature is established by search, not by any systematic review — a negative (counterexample) answer could exist in sources I did not reach.

research/AMR-010-0110.md

@@ -0,0 +1,71 @@
+---
+id: AMR-010-0110
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+# AMR-010-0110 — Canary's "power-full subgroups" question for hyperbolic groups
+
+## Problem (corrected statement if needed)
+
+Statement verified verbatim against the source PDF
+([Bestvina, *Questions in Geometric Group Theory*, updated July 2004](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), Q 1.10 — no update note is attached to this question in the list):
+
+> **Q 1.10 (Canary).** Let $G$ be word-hyperbolic and $H$ a finitely presented subgroup of $G$.
+> Suppose that for every $g\in G$ there is $n>0$ such that $g^n\in H$. Does it follow that $H$ has finite index in $G$?
+> *(Bestvina's note: "Yes if $H$ is quasi-convex, since then $\Lambda(H)=\Lambda(G)$.")*
+
+The dataset wording was accurate; no correction was needed. Write $\sqrt{H}:=\{g\in G : \exists n>0,\ g^n\in H\}$; the hypothesis is $\sqrt{H}=G$.
+
+## Status / Literature
+
+- I found **no published solution**. Multiple search angles (Canary + hyperbolic + power + finite index; Bestvina problem list status; arXiv API; DuckDuckGo/Bing mirrors) turned up nothing resolving the question. Caveat: the search tools were heavily rate-limited during this session, so the sweep was shallower than intended; I am not aware of any resolution from my own knowledge of the literature either. As far as I can tell the problem is **open**.
+- The obstruction to a counterexample is genuinely famous. If $H\trianglelefteq G$ is a *normal* counterexample, then $Q=G/H$ is an infinite **torsion** group (the hypothesis says every element of $G$ has a power in $H$, i.e. every element of $Q$ has finite order), and $Q$ is **finitely presented**: $G$ is finitely presented and $H$ is finitely generated (being finitely presented), so adding a finite generating set of $H$ as relators to a finite presentation of $G$ gives a finite presentation of $Q$. Hence a normal counterexample would produce an **infinite finitely presented torsion group**, whose existence is a notorious open problem (see e.g. the MathOverflow discussion
+  [“An infinite torsion group $G$ with finite type $K(G,1)$?”](https://mathoverflow.net/questions/239057/), which calls the existence of an infinite finitely presented torsion group “a famous open problem”; also listed in the Kourovka Notebook). The general belief is that such groups exist, but even then one would need one *as a quotient of a hyperbolic group with finitely presented kernel*, which is much stronger (see “What remains”).
+- Infinite torsion *quotients* of hyperbolic groups certainly exist (Olshanskii: $G/G^n$ is infinite for large odd $n$ when $G$ is non-elementary hyperbolic), but the kernel $G^n$ is not finitely presented (typically not even finitely generated relative to its normal structure in a controllable way), so this does not touch the question.
+- Positive territory already in the literature (used below): the quasi-convex case (noted in the source itself); limit groups are locally quasi-convex (Wilton, *Hall's theorem for limit groups*, GAFA 2008); Canary's covering theorem for hyperbolic 3-manifolds (R. Canary, *A covering theorem for hyperbolic 3-manifolds and its applications*, Topology 1996), which says a finitely generated subgroup of (a finite extension of) a closed hyperbolic 3-manifold group is either geometrically finite (quasi-convex) or a virtual fiber.
+
+## Work done
+
+I verified the statement against the source PDF (extracted the text of the July 2004 list and confirmed Q 1.10 carries no “Update”), surveyed the status as above, and proved the following partial results and reductions. All arguments below are my own derivations from standard facts.
+
+**Proposition 1 (full limit set).** If $\sqrt{H}=G$ with $G$ non-elementary hyperbolic and $H\le G$ arbitrary, then $\Lambda H=\partial G$.
+*Proof.* Fixed point pairs of loxodromic elements are dense in $\partial G\times\partial G$; if $g$ is loxodromic and $g^n\in H$, then $g^n$ is loxodromic in $H$ with the same fixed points $g^{\pm\infty}$, so $g^{\pm\infty}\in\Lambda H$. Since $\Lambda H$ is closed and contains a dense subset of $\partial G$, $\Lambda H=\partial G$. $\square$
+
+**Corollary 2 (quasi-convex case — Bestvina's remark).** If $H$ is quasi-convex and $\sqrt{H}=G$, then $[G:H]<\infty$.
+*Proof.* By Proposition 1, $\Lambda H=\partial G$. A quasi-convex subgroup of infinite index in a hyperbolic group has nowhere-dense limit set in $\partial G$ (standard: the orbit $H\!\cdot\!x$ misses a uniform neighborhood of a conical limit point of $G$ lying outside $\Lambda H$; such points exist because $\Lambda H\neq\partial G$ for infinite-index quasi-convex $H$). Hence $[G:H]<\infty$. $\square$
+
+**Proposition 3 (the normal case is exactly a torsion-quotient problem).** For $H\trianglelefteq G$ ($H$ f.p.):
+$$\sqrt{H}=G \ \Longleftrightarrow\ G/H\text{ is a torsion group},$$
+and if in addition $[G:H]=\infty$ then $G/H$ is an **infinite finitely presented torsion group**. Consequently:
+- If no infinite finitely presented torsion group exists (a well-known conjectural answer to a famous open problem), then Canary's question has answer **yes** for every normal $H$.
+- The argument needs only $H$ *finitely generated*: any normal counterexample with $H$ f.g. (a fortiori f.p.) yields an infinite f.p. torsion group. So even the f.g. analogue of the normal case is exactly as hard as the famous problem.
+
+**Proposition 4 (Rips obstruction — counterexamples cannot be built cheaply).** The Rips construction gives, for any f.p. group $Q$, a short exact sequence $1\to K\to G\to Q\to1$ with $G$ hyperbolic (small-cancellation) and $K$ finitely generated (2-generated). Hence: *if* an infinite f.p. torsion group $Q$ exists, the f.g. version of Canary's question has a negative answer. But the Rips kernel $K$ is not known (and not expected) to be finitely presented, so this does not refute the question as stated. Upgrading the kernel to f.p. via fiber-product machinery (Baumslag–Bridson–Miller–Short “1-2-3 theorem” style) would require $Q$ of type $F_3$, i.e. an infinite torsion group of type $F_3$ — strictly harder than the famous open problem (and its existence is likewise open; cf. the MathOverflow thread above, which asks exactly about torsion groups with strong finiteness properties).
+
+**Proposition 5 (almost malnormal case).** Suppose $G$ is torsion-free hyperbolic, $H\le G$ is almost malnormal, and $\sqrt{H}=G$. Then $H=G$.
+*Proof.* Suppose $g\notin H$ with $g^n\in H$, $n>1$. Then $g^n = g(g^n)g^{-1}\in H\cap gHg^{-1}$, and $g^n$ has infinite order ($G$ torsion-free), so $H\cap gHg^{-1}$ is infinite with $g\notin H$, contradicting almost malnormality. Hence no such $g$ exists, i.e. $\sqrt{H}=G$ forces $H=G$. $\square$
+(With torsion allowed, the same argument works unless every offending power $g^n$ has finite order.)
+
+**Proposition 6 (closed hyperbolic 3-manifold groups — yes, even for f.g. $H$).** Let $G=\pi_1(M)$ with $M$ a closed hyperbolic 3-manifold (or any torsion-free convex-cocompact Kleinian group), and let $H\le G$ be finitely generated with $\sqrt{H}=G$. Then $[G:H]<\infty$.
+*Proof.* By Canary's covering theorem (the ambient group is topologically tame, being convex cocompact), $H$ is either geometrically finite or a *virtual fiber*: in the latter case a finite-index subgroup of $H$ is the fiber kernel of a fibration of a finite cover $M'\to S^1$, so $H$ has a quotient surjecting onto $\mathbb{Z}$ (up to finite kernel). Then there is $g\in G$ whose image in that $\mathbb{Z}$-quotient has infinite order, and no power $g^n$ ($n>0$) lies in $H$ — contradicting $\sqrt{H}=G$. So $H$ is geometrically finite, hence quasi-convex in the hyperbolic group $G$ (for closed/convex-cocompact hyperbolic 3-manifold groups, geometric finiteness = quasi-convexity). Now apply Corollary 2. $\square$
+
+**Corollary 7 (locally quasi-convex groups).** If $G$ is hyperbolic and every f.g. subgroup is quasi-convex (e.g. free groups, closed surface groups, and more generally limit groups by Wilton's theorem), then the answer is **yes** for every f.g. $H$, since $H$ is quasi-convex and Corollary 2 applies.
+
+**Why the general case is hard (failed-attempt analysis).** Finitely presented subgroups of hyperbolic groups can be extremely distorted: Brady (1999) constructed hyperbolic groups containing f.p. subgroups that are not hyperbolic (not quasi-convex, wildly distorted). So no intrinsic geometry of $H$ is available; the only leverage is the algebraic power condition. The two natural attacks both hit famous walls:
+1. *Counterexample route* — blocked by Propositions 3–4: one must first produce an infinite f.p. torsion group (open since Novikov–Adian, cf. the MO thread), and then realize it as a quotient of a hyperbolic group with f.p. kernel (apparently harder).
+2. *Proof route* — the hypothesis gives $\Lambda H=\partial G$ (Proposition 1), and the quasi-convex conclusion would follow from the statement “a f.p. subgroup of a hyperbolic group with full limit set has finite index”; but finitely presented subgroups need not have well-behaved limit-set dynamics (they need not be hyperbolic), and I know of no theorem that promotes “f.p. + full limit set” to finite index. Residual-finiteness arguments fail: proper power-dense subgroups of finite groups exist (e.g. $2\mathbb{Z}/4\subset\mathbb{Z}/4$), so even LERF does not obviously separate a hypothetical $g\notin H$.
+
+## Result
+
+The problem appears **open**; I could not find any published resolution. Rigorous partial progress obtained here:
+
+- **Reduction of the normal case:** for $H\trianglelefteq G$ the question is equivalent to “does a hyperbolic group admit an infinite finitely presented torsion quotient?”, and any counterexample (even with $H$ merely finitely generated) would solve the famous open problem on the existence of infinite finitely presented torsion groups (Propositions 3–4).
+- **Proved special cases:** the answer is *yes* when $H$ is quasi-convex (Corollary 2, Bestvina's remark made precise via Proposition 1); when $H$ is almost malnormal and $G$ is torsion-free (Proposition 5); when $G$ is a closed hyperbolic 3-manifold group — for every f.g. $H$, via Canary's covering theorem (Proposition 6); and when $G$ is locally quasi-convex, e.g. a limit group (Corollary 7).
+- **Structural consequence:** any $H$ with $\sqrt H = G$ satisfies $\Lambda H=\partial G$ (Proposition 1), so the question is a strengthening of the (also delicate) question whether f.p. subgroups with full limit set have finite index.
+
+## What remains
+
+- The general case: $H$ f.p., non-normal, badly distorted. Nothing seems to be known here beyond the cases above.
+- The normal case is pinned to a notorious problem: decide whether infinite f.p. torsion groups exist, and more specifically whether one can be a quotient of a hyperbolic group with f.p. (or type-$F_3$) kernel. A “no” to the latter settles Canary's normal case affirmatively; a “yes” with f.p. kernel settles Canary's question negatively.
+- A proof route might try to show directly that “f.p. + $\sqrt{H}=G$” forces quasi-convexity of $H$ (which would suffice by Corollary 2), but no current technique (JSJ, combination theorems, cubulation) seems to touch distorted f.p. subgroups without extra hypotheses.
+- Literature follow-up when search tools are not rate-limited: check whether Canary himself, or authors citing Bestvina's list (e.g. via Google Scholar citations of the list), have recorded progress on Q 1.10, and whether the term “power-full/radically dense subgroup” has appeared in print for this property.

research/AMR-010-0111.md

@@ -0,0 +1,123 @@
+---
+id: AMR-010-0111
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0111 — Whyte's question: can every infinite-index subgroup of a 1-ended hyperbolic group be free?
+
+## Problem (corrected statement if needed)
+
+Source: Bestvina's problem list "Questions in Geometric Group Theory" (updated July 2004),
+Question 1.11, author-hosted PDF at <https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>.
+The PDF was fetched and the wording checked verbatim; the dataset transcription is **exact**:
+
+> **Q 1.11. (Whyte)** Let Γ be a 1-ended hyperbolic group which is not virtually a surface group.
+> Can every infinite index subgroup be free?
+
+No correction needed. Reading of the question (confirmed by its position in the list, directly after
+Gromov's Q 1.6/1.7 on surface subgroups and the "opposite possibility" that all infinite-index
+subgroups might be free): *does there exist* a 1-ended hyperbolic group Γ, not virtually a surface
+group, in which every infinite-index subgroup is free? A **negative** answer means every such Γ
+contains a non-free subgroup of infinite index. The exclusion of virtual surface groups is essential:
+every infinite-degree cover of a closed hyperbolic surface is a non-compact surface, so every
+infinite-index subgroup of a surface group is free — surface groups are the motivating (excluded)
+examples. Free groups F_n (n ≥ 2) also have all subgroups free but are infinitely ended, hence
+excluded by 1-endedness.
+
+## Status / Literature
+
+All items below were verified against the arXiv API or Crossref (DOIs resolve and metadata matches).
+
+- **H. Wilton, "Surface groups among cubulated hyperbolic and one-relator groups",
+  arXiv:2406.02121** (v1 June 2024; v3, January 2026, "final version accepted for publication"
+  per the arXiv comment). Theorem A: *a cubulated hyperbolic group G, unless free or a surface
+  group, has a one-ended quasiconvex subgroup of infinite index.* A one-ended subgroup is non-free,
+  so this answers Whyte's Q 1.11 (and Gromov's Q 1.7) **negatively for all cubulated hyperbolic
+  groups** — the paper's abstract says exactly this ("answering questions of Gromov and Whyte (in a
+  special case) and Wise", referencing Bestvina's Questions 1.7 and 1.11). Consequences: C'(1/6)
+  small-cancellation groups and, via virtual specialness (Agol), all hyperbolic 3-manifold groups.
+  A similar statement is proved for one-relator groups. The paper itself notes that removing the
+  cubulation hypothesis "seems to be well beyond current technology" — i.e., the general case is open.
+
+- **H. Wilton, "One-ended subgroups of graphs of free groups with cyclic edge groups",
+  arXiv:1102.2866; Geom. Topol. 16 (2012), 665–683, DOI 10.2140/gt.2012.16.665.**
+  A one-ended hyperbolic group that is the fundamental group of a graph of free groups with cyclic
+  edge groups is either a surface group or contains a finitely generated one-ended subgroup of
+  infinite index. Same conclusion for limit groups. Hence Whyte's question has a negative answer
+  for these classes.
+
+- **J. Kahn and V. Markovic, "Immersing almost geodesic surfaces in a closed hyperbolic three
+  manifold", Ann. of Math. 175 (2012), 1127–1190, DOI 10.4007/annals.2012.175.3.4** (Crossref
+  verified). Every closed hyperbolic 3-manifold group contains a (quasi-Fuchsian) closed surface
+  subgroup of genus ≥ 2 — an infinite-index, non-free subgroup. Negative answer for closed
+  hyperbolic 3-manifold groups (this is the surface subgroup conjecture for that class).
+
+- **U. Hamenstädt, "Incompressible surfaces in rank one locally symmetric spaces",
+  Geom. Funct. Anal. 25 (2015), 815–859, DOI 10.1007/s00039-015-0330-y** (Crossref verified).
+  Cocompact lattices in rank-one simple Lie groups contain quasi-Fuchsian surface subgroups;
+  negative answer for those lattices.
+
+- **I. Agol, "The virtual Haken conjecture", Doc. Math. 18 (2013), 1045–1087, DOI 10.4171/dm/421**
+  (Crossref verified; with appendix by Agol–Groves–Manning). Cubulated hyperbolic groups are
+  virtually special; combined with the cubulation of hyperbolic 3-manifold groups this shows the
+  scope of "cubulated" in Wilton's Theorem A is very large.
+
+Context: Gromov's surface-subgroup question (Bestvina's Q 1.6 — does every 1-ended hyperbolic group
+contain a closed surface subgroup?) is still **open in general**. A positive answer to Gromov's
+conjecture would give a negative answer to Whyte's question (a closed surface subgroup of genus ≥ 2
+is non-free, and has infinite index since Γ is not virtually a surface group). Whyte's question is
+strictly weaker: a hypothetical "yes" example would be a counterexample to Gromov's conjecture, but
+a negative answer to Whyte does not require surface subgroups, only non-free ones.
+
+## Work done
+
+- Fetched Bestvina's `questions-updated.pdf` and confirmed the wording of Q 1.11 verbatim
+  (dataset transcription is accurate, including attribution "(Whyte)"; the adjacent Q 1.12 is a
+  different Whyte question about co-Hopfianity).
+- Verified every citation above against the arXiv API (`export.arxiv.org/api/query`) or Crossref
+  (`api.crossref.org/works/<DOI>`). One guessed DOI for Hamenstädt's paper returned 404; the correct
+  DOI (10.1007/s00039-015-0330-y) was found via a Crossref bibliographic query and used instead.
+- Mathematical reasoning (standard consequences of known theorems; no computation): suppose Γ is a
+  counterexample to the negative answer, i.e. 1-ended hyperbolic, not virtually a surface group,
+  with all infinite-index subgroups free. Then Γ cannot split over ℤ: in a splitting of a 1-ended
+  hyperbolic group over a 2-ended subgroup the vertex groups are quasiconvex (Bowditch's JSJ/cut-point
+  theory, Acta Math. 180 (1998)) and of infinite index, hence would be free; Γ would then be a graph
+  of free groups with cyclic edge groups, and Wilton's 2012 theorem forces Γ to be a surface group or
+  to contain a one-ended infinite-index subgroup — contradiction. So any counterexample must be
+  JSJ-rigid (no splitting over 2-ended subgroups), equivalently its Gromov boundary has no local cut
+  points (Bowditch), and by Wilton's 2024 theorem it cannot be cubulated. This reduction to the
+  rigid, non-cubulated case is essentially the strategy of Wilton's papers; the remaining case is
+  precisely where current techniques (which all proceed by finding splittings or cube actions) fail.
+
+## Result
+
+The question is **open in full generality**, but the answer is now known to be **"no"** (a non-free
+infinite-index subgroup always exists) for every major class where the question has been attacked:
+
+- cubulated hyperbolic groups (Wilton 2024, arXiv:2406.02121, accepted for publication) — including
+  C'(1/6) small-cancellation groups and, via Agol–Wise, hyperbolic 3-manifold groups;
+- one-relator groups (same paper);
+- hyperbolic graphs of free groups with cyclic edge groups, and limit groups (Wilton 2012,
+  DOI 10.2140/gt.2012.16.665);
+- closed hyperbolic 3-manifold groups and rank-one lattices, where even surface subgroups exist
+  (Kahn–Markovic, DOI 10.4007/annals.2012.175.3.4; Hamenstädt, DOI 10.1007/s00039-015-0330-y).
+
+Moreover, any would-be positive example must simultaneously be a counterexample to Gromov's
+surface-subgroup conjecture, admit no splitting over ℤ, have a boundary without local cut points,
+and admit no proper cocompact action on a CAT(0) cube complex.
+
+## What remains
+
+- The general case: 1-ended hyperbolic groups not known to be cubulated (e.g., generic
+  random/Gromov-model hyperbolic groups, for which cubulation fails or is unknown). Nothing
+  currently rules out an example with all infinite-index subgroups free.
+- Even for cubulated groups, the stronger Gromov question (a genuine *surface* subgroup, not just a
+  one-ended quasiconvex one) is open in general — Wilton's Theorem A produces one-ended subgroups,
+  not surface groups.
+- Obstacle: all known methods locate non-free subgroups via splittings over ℤ or via cubical
+  geometry; JSJ-rigid non-cubulated groups are beyond these techniques (Wilton's own remark).
+- Natural next steps: settle the question for random hyperbolic groups at various densities; decide
+  whether one-endedness of a quasiconvex subgroup can be promoted to a surface subgroup in the
+  cubulated case; boundary-based approaches (existence of a topological circle in ∂Γ would, modulo
+  Cannon-type conjectures, yield surface subgroups).

research/AMR-010-0112.md

@@ -0,0 +1,61 @@
+---
+id: AMR-010-0112
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0112 — Whyte's question: finite-index subgroups isomorphic to infinite-index subgroups in 1-ended hyperbolic groups
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is faithful to the source. The problem is Question 1.12 in Mladen Bestvina's open-problem list "Questions in Geometric Group Theory" (major revision August 22, 2000; updated version at https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), attributed to Kevin Whyte:
+
+> (Whyte) Let Γ be a 1-ended hyperbolic group. Can a finite index subgroup of Γ be isomorphic to a subgroup of Γ of infinite index?
+
+No correction needed. Note the attribution caveat in Bestvina's list: "Names in parentheses reflect the person I heard the question from." The same question was also asked by Kapovich (I. Kapovich, "Arithmetic aspects of self-similar groups", Groups Geom. Dyn. 6 (2012), DOI: 10.4171/GGD/172, Section 2, where the property of admitting no such pair is called "weakly coHopfian").
+
+## Status / Literature
+
+**Answer: YES — such groups exist. The question is fully resolved (affirmatively) by Stark–Woodhouse (2021).**
+
+Verified citations:
+
+1. **E. Stark, D. J. Woodhouse, "Hyperbolic Groups That Are Not Commensurably Co-Hopfian"**, International Mathematics Research Notices (IMRN) 2021, no. 1, 579–595. DOI: 10.1093/imrn/rnaa033 (verified via Crossref API); arXiv:1812.07799 (verified via arXiv API; v3, 2020).
+   - A group Γ is *commensurably coHopfian* if no finite-index subgroup of Γ is isomorphic to an infinite-index subgroup of Γ. Whyte's question asks whether every 1-ended hyperbolic group is commensurably coHopfian. The paper explicitly states it answers Whyte's Question 1.12 from Bestvina's list.
+   - **Theorem 1.1:** There exist one-ended hyperbolic groups that are not commensurably coHopfian. Main example: a *simple surface amalgam* X built from three genus-one surfaces with one boundary component, boundaries identified. They construct a degree-3 cover X₁ → X (each surface covered by a genus-2 one-boundary-component surface) and a degree-4 cover X₂ → X, with a π₁-injective proper embedding X₁ ↪ X₂ (a retraction). Then π₁(X₁) ≅ π₁(X₂) sit inside π₁(X): the former has finite index (degree-4 cover), the latter infinite index, yet π₁(X₁) embeds in π₁(X₂), so π₁(X) contains a finite-index subgroup isomorphic to an infinite-index subgroup.
+   - **Theorem 1.2:** The fundamental group of *every* simple surface amalgam (union of k ≥ 3 negative-Euler-characteristic one-boundary-component surfaces with boundaries identified) is not commensurably coHopfian. These groups are one-ended and hyperbolic (Bestvina–Feighn combination theorem; they even admit CAT(-1) metrics).
+   - Context within the paper: Sela proved every torsion-free one-ended hyperbolic group is coHopfian (Moioli's thesis extended this to all one-ended hyperbolic groups), so the answer to Whyte's question was genuinely uncertain; Strebel (Comment. Math. Helv. 52 (1977), DOI: 10.1007/BF02567371) proved infinite-index subgroups of Poincaré duality groups have strictly smaller cohomological dimension, hence PD groups (e.g. closed hyperbolic manifold groups) ARE commensurably coHopfian — so the answer is "yes in general, no for PD groups".
+   - The constructed infinite-index embeddings are retractions, hence quasi-isometric embeddings that are not quasi-isometries; these are also the first known examples of one-ended hyperbolic groups that are not quasi-isometrically coHopfian.
+   - The paper poses Conjecture 1.3: failure of commensurable coHopficity for a one-ended hyperbolic group should be tied to the presence of maximal hanging Fuchsian vertex groups in the Bowditch JSJ decomposition.
+
+2. **N. Lazarovich, "Finite index rigidity of hyperbolic groups"**, arXiv:2302.04484 (v3, 2024; verified via arXiv API).
+   - Proves that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index. This settles (negatively) the closely related follow-up question recorded as Question 1.6 in the Stark–Woodhouse paper (attributed to Bestvina): no one-ended hyperbolic group contains isomorphic finite-index subgroups of *different* indices. It complements Stark–Woodhouse: the finite-index/infinite-index phenomenon cannot occur between two finite-index subgroups.
+
+## Work done
+
+- Read the dataset item and identified the source as Bestvina's "Questions in Geometric Group Theory", Q1.12 (Whyte). The wording matches the author's PDF verbatim, so no correction was needed.
+- Web-searched the question; located the Oxford ORA preprint and the arXiv listing of the Stark–Woodhouse paper, which explicitly states it answers Whyte's Question 1.12 on Bestvina's list.
+- Fetched and read the Stark–Woodhouse preprint (introduction and the main construction in Sections 2–3) to confirm exactly what is proved and how.
+- Verified the publication record via Crossref (DOI 10.1093/imrn/rnaa033, IMRN 2021(1), 579–595) and the arXiv API (arXiv:1812.07799; Lazarovich arXiv:2302.04484). All citations in this report were verified to exist through one of these two APIs or the Bestvina PDF itself.
+- Reproduced the logical structure of the main example independently (see Result) — the construction is elementary (covering-space theory plus Euler characteristic bookkeeping via Neumann's Lemma 3.2 in Algebr. Geom. Topol. 1 (2001), DOI: 10.2140/agt.2001.1.411).
+
+## Result
+
+The question is solved in the literature with answer **yes**: there exist one-ended hyperbolic groups Γ containing a finite-index subgroup H ≤ Γ and an infinite-index subgroup K ≤ Γ with H ≅ K.
+
+Sketch of the Stark–Woodhouse main example (my summary of their §2): Let X = Σ₁ ∪ Σ₂ ∪ Σ₃ where each Σᵢ is a genus-1 surface with one boundary circle, all boundaries glued to a single S¹. Then π₁(X) is a one-ended hyperbolic group (Bestvina–Feighn, since each π₁(Σᵢ) is free amalgamated along a malnormal cyclic subgroup).
+
+- *Degree-3 cover X₁:* by Neumann's covering lemma, each Σᵢ has a 3-sheeted cover with exactly one boundary component; its Euler characteristic is 3·(−1) = −3, so it is a genus-2 surface with one boundary. Gluing gives a degree-3 cover X₁ → X of the same "simple surface amalgam" form.
+- *Degree-4 cover X₂:* each Σᵢ has a 2-sheeted cover Σᵢ″ with two boundary components (still genus 1). Glue one boundary component of each Σᵢ″ to form one amalgam circle, and attach extra copies of the Σⱼ along the other boundary components; this gives a degree-4 cover X₂ → X.
+- *Embedding:* X₁ embeds π₁-injectively as a proper sub-amalgam of X₂ (visibly a retract of X₂), so π₁(X₁) ≅ π₁(X₂) appears inside π₁(X₂) as an infinite-index subgroup. Since both π₁(X₁) (index 3) and π₁(X₂) (index 4) are finite-index in π₁(X), the group Γ = π₁(X) contains a finite-index subgroup (π₁(X₁), via index 3) isomorphic to an infinite-index subgroup (π₁(X₁) ⊂ π₁(X₂) ⊂ Γ). ∎
+
+Their Theorem 3.1 extends this to all simple surface amalgams by solving a linear system in covering degrees (their Claim 3.2) to build two finite covers X′, X″ with X′ embedding π₁-injectively in X″.
+
+The phenomenon is genuinely new relative to classical rigidity: one-ended hyperbolic groups are coHopfian (Sela; Moioli), and Poincaré duality groups (e.g. closed hyperbolic manifold groups) are commensurably coHopfian by Strebel's cohomological-dimension argument — so Whyte's question has answer "yes" in general but "no" for important subclasses.
+
+## What remains
+
+- **Characterization problem (Stark–Woodhouse Conjecture 1.3, still open to my knowledge):** for a one-ended hyperbolic group, is failure of commensurable coHopficity equivalent to the presence of maximal hanging Fuchsian vertex groups in the Bowditch JSJ decomposition? Stark–Woodhouse give supporting examples on both sides (mixed JSJ examples that are and are not commensurably coHopfian) but the general conjecture is open; they caution that highly distorted (non-quasiconvex) embeddings may require a quasiconvexity hypothesis.
+- **Quasi-isometric coHopficity:** the embeddings constructed are retractions, hence the first examples of one-ended hyperbolic groups failing to be quasi-isometrically coHopfian. Classifying which one-ended hyperbolic groups are QI-coHopfian remains open (related work: Kapovich–Lukyanenko for non-uniform rank-one lattices, DOI: 10.1090/S1088-4173-2012-00246-9).
+- **Different indices, both finite:** Lazarovich (arXiv:2302.04484) closed the variant asking for isomorphic finite-index subgroups of *different* indices — impossible for non-elementary hyperbolic groups.
+- Whether the commensurable-coHopficity dichotomy can be detected from the Gromov boundary or conformal dimension appears unexplored.

research/AMR-010-0113.md

@@ -0,0 +1,58 @@
+---
+id: AMR-010-0113
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0113 — Combination theorem for relatively hyperbolic groups (Swarup)
+
+## Problem (corrected statement if needed)
+
+Source: M. Bestvina, "Questions in Geometric Group Theory" (major revision August 2000, updated July 2004), Question 1.13 (PDF page 3), author-hosted PDF at https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf .
+
+Original wording, verified against the source PDF:
+
+> **Q 1.13. (Swarup)** Prove the combination theorem for relatively hyperbolic groups.
+
+The July-2004 update in the list itself already records the solution:
+
+> "Update: Dahmani [Dah03] and Alibegović [Ali] have versions adapted for use to limit groups."
+
+The dataset transcription matches the source exactly; no correction needed. The question asks for the relative analogue of the Bestvina–Feighn Combination Theorem for hyperbolic groups (M. Bestvina and M. Feighn, "A combination theorem for negatively curved groups", *J. Differential Geom.* 35 (1992), 85–101, DOI 10.4310/jdg/1214447806).
+
+## Status / Literature
+
+**Solved in the literature, by three complementary theorems** (all citations verified via Crossref / arXiv):
+
+1. **F. Dahmani, "Combination of convergence groups", *Geometry & Topology* 7 (2003), 933–963.** DOI 10.2140/gt.2003.7.933 (verified on Crossref). A *dynamical* combination theorem: he shows that suitable amalgamated products / HNN extensions of relatively hyperbolic groups (viewed as convergence groups) are again relatively hyperbolic, with the expected peripheral structure. This was the first published combination theorem for relative hyperbolicity and was tailored to applications to limit groups — it is the key tool proving that limit groups are relatively hyperbolic with respect to their maximal abelian subgroups of rank ≥ 2 (Q 3.8 of the same list).
+
+2. **E. Alibegović, "A combination theorem for relatively hyperbolic groups", *Bulletin of the London Mathematical Society* 37(3) (2005), 459–466.** DOI 10.1112/S0024609304004059 (verified on Crossref); arXiv:math/0310257 (abstract page verified). A Bestvina–Feighn-style combination theorem for amalgams of relatively hyperbolic groups along "liminal" edge groups satisfying malnormality-type conditions, again with applications to limit groups.
+
+3. **M. Mj and L. Reeves, "A combination theorem for strong relative hyperbolicity", *Geometry & Topology* 12(3) (2008), 1777–1798.** DOI 10.2140/gt.2008.12.1777 (verified via Crossref reference lists of later papers and the Project Euclid page); arXiv:math/0611601 (abstract verified — the abstract states explicitly: "This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and **answers a question of Swarup**."). A *geometric* combination theorem for trees of (strongly) relatively hyperbolic metric spaces, with conditions different from those of Dahmani and Alibegović, plus a converse to the main theorem.
+
+Later refinements/variants also exist, e.g. R. Tomar, "A combination theorem for relatively acylindrical graphs of relatively hyperbolic groups", *Topology Appl.* 380 (2026), 109692, DOI 10.1016/j.topol.2025.109692 (verified on Crossref), and an unpublished algebraic version for 2-complexes of relatively hyperbolic groups by F. Gautero ("An algebraic combination theorem for graphs of relatively hyperbolic groups", preprint, 2011, author-hosted; not formally published, cited but not verified as refereed).
+
+## Work done
+
+- Read `worklist/AMR-010-0113.md`; fetched the Bestvina source PDF and confirmed the exact wording of Q 1.13 and the July-2004 update line.
+- Web-searched for the resolution; identified the three main papers above.
+- Verified each citation against Crossref (`api.crossref.org/works/...`): Dahmani (DOI 10.2140/gt.2003.7.933 — full record including title, journal, volume, pages), Alibegović (DOI 10.1112/S0024609304004059 — full record), Mj–Reeves (DOI 10.2140/gt.2008.12.1777 — confirmed via Project Euclid listing and via the Crossref-deposited reference lists of Krishna, *Proc. Math. Sci.* 130 (2020), and Tomar 2026), Bestvina–Feighn (DOI 10.4310/jdg/1214447806 — confirmed in Crossref reference lists). arXiv abstract pages for math/0611601 and math/0310257 were fetched and confirm titles/authors.
+- No independent new mathematics was attempted: the question is fully settled in the published literature, so the appropriate output is a rigorous triage (a "solve-by-you" attempt at an L3 problem already solved by three major papers would add nothing).
+
+## Result
+
+Swarup's question is **answered affirmatively in the literature**, in three distinct frameworks that mirror the different definitions of relative hyperbolicity:
+
+- **Dynamical/convergence-group version (Dahmani 2003).** If groups acting as convergence groups (relatively hyperbolic) are combined along parabolic-type ("liminal") subgroups satisfying geometric-finiteness and intersection-control hypotheses, the amalgamated product/HNN extension acts as a convergence group on a suitably assembled compactum and is relatively hyperbolic relative to the expected peripherals. Applied to show limit groups are relatively hyperbolic w.r.t. maximal noncyclic abelian subgroups.
+- **Amalgam version (Alibegović 2005).** For a one-edge graph of relatively hyperbolic groups with liminal edge group satisfying an almost-malnormality condition and a compatibility ("isolated"-type) condition on the peripherals, the fundamental group of the graph of groups is relatively hyperbolic relative to the images of the vertex peripherals not contained in the edge group.
+- **Geometric version (Mj–Reeves 2008), the one explicitly billed as answering Swarup's question.** For a tree of strongly relatively hyperbolic metric spaces satisfying (i) the qi-embedded condition (edge spaces quasi-isometrically embed into vertex spaces with edge-to-vertex qi-embeddings), (ii) the strictly type-preserving condition (peripheral/horosphere-like sets map to peripheral sets), and (iii) a uniform hallway-flare condition (the relative analogue of the Bestvina–Feighn flare condition), the total space is strongly relatively hyperbolic relative to the natural family of horosphere-like subsets; a **weak** combination theorem (electrocution/electric-space hyperbolicity) holds under (i)+(ii) with a milder flare condition, and they prove a **converse**: strong relative hyperbolicity of the total space forces the qi-embedded condition.
+
+Together these subsume the classical Bestvina–Feighn theorem (recover it by taking all peripherals trivial/hyperbolic) and establish the general combination principle Swarup asked for.
+
+## What remains
+
+The original question is closed. Remaining activity is in refinements rather than in the problem itself:
+
+- Combination theorems under weaker hypotheses (e.g. relatively acylindrical splittings; Tomar 2026; Pal–Tomar work on finite relative height of splittings, arXiv:2207.03167 — not fully verified here).
+- A fully published algebraic combination theorem for general 2-complexes of relatively hyperbolic groups (Gautero's version remains a preprint).
+- Companion questions: Cannon–Thurston maps and limit-set intersection theorems for the combined relatively hyperbolic group (partially answered by Mj–Pal, Sardar, Krishna).

research/AMR-010-0115.md

@@ -0,0 +1,48 @@
+---
+id: AMR-010-0115
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0115 — Is every word-hyperbolic group residually finite?
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is verbatim correct. In Bestvina, *Questions in Geometric Group Theory* (updated July 2004), §1.4 "Residual Finiteness", Question 1.15 reads:
+
+> **Q 1.15.** Is every word-hyperbolic group residually finite?
+
+Source: [Bestvina's problem list (author PDF)](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), fetched and confirmed 2026-08-04. The source appends three notes:
+
+- (a) D. Wise constructed a finite 2-dimensional locally CAT(0) complex whose fundamental group is **not** residually finite (so the naive CAT(0) analogue has a negative answer; the hyperbolic case is genuinely sharper).
+- (b) Z. Sela showed torsion-free word-hyperbolic groups are Hopfian (a weak property implied by residual finiteness).
+- (c) M. Kapovich observed that **non-linear** word-hyperbolic groups exist (quotient a uniform lattice in quaternionic hyperbolic space by a "random" relation; super-rigidity forces every linear representation to be faithful or have finite image). Hence Malcev's theorem (finitely generated linear ⇒ residually finite) cannot settle the question.
+
+## Status / Literature
+
+**Open**, and one of the central open problems of geometric group theory; it goes back to Gromov (1987), who suggested the answer might be negative ("probably 'generic' word-hyperbolic groups admit no sequences of subgroups of finite index with trivial intersection", [Gro87, §5.3.B]). All citations below were verified against Crossref/arXiv metadata during this review:
+
+- **Kapovich–Wise equivalence.** I. Kapovich and D. T. Wise, *The equivalence of some residual properties of word-hyperbolic groups*, J. Algebra 223 (2000), no. 2, 562–583, DOI [10.1006/jabr.1999.8104](https://api.crossref.org/works/10.1006/jabr.1999.8104) (verified: authors, title, journal, volume, pages all match). They prove that "every hyperbolic group is residually finite" is equivalent to "every hyperbolic group has a proper finite-index subgroup", and that if any hyperbolic group fails residual finiteness then there exists a non-elementary hyperbolic group with **no nontrivial finite quotients at all**. Olshanskii independently made the same observation.
+- **Agol–Groves–Manning.** I. Agol, D. Groves, J. F. Manning, *Residual finiteness, QCERF and fillings of hyperbolic groups*, Geom. Topol. 13 (2009), 1043–1073, DOI [10.2140/gt.2009.13.1043](https://api.crossref.org/works/10.2140/gt.2009.13.1043) (verified). Main theorem: if every hyperbolic group is residually finite, then every hyperbolic group is **QCERF** (all quasi-convex subgroups are separable). Combined with Wise's quasi-convex hierarchy theorem, this gives: residual finiteness of all hyperbolic groups ⇒ every *cubulated* hyperbolic group is virtually special. So a positive answer would have consequences as strong as virtual specialness — evidence that the problem is at least as deep as the full virtual-specialness package proved in the 3-manifold case (Agol, Wise, 2012–2013).
+- **Haglund–Wise** *Special cube complexes*, GAFA 17 (2008), DOI 10.1007/s00039-007-0629-4 (verified indirectly: it appears as a Crossref-deposited reference "HW08" inside the Agol–Groves–Manning record above): virtually special groups embed in right-angled Artin groups, hence are linear over ℤ, hence residually finite. This is the engine behind all known positive cases.
+- **Sela's Hopficity** (note (b) in the source): Z. Sela, *Endomorphisms of hyperbolic groups I: The Hopf property*, Topology 38 (1999), 301–321, DOI 10.1016/S0040-9383(98)00015-9 (verified indirectly as a Crossref-deposited reference in the Kapovich–Wise record). Residually finite f.g. groups are Hopfian, so this is consistent with, but far weaker than, residual finiteness.
+- **Current-status confirmation.** A 2019 survey (Bou-Rabee, *Finite and infinite quotients of discrete and indiscrete groups*, [arXiv:1709.05949](https://arxiv.org/pdf/1709.05949v2)) treats it as "a major open problem"; a 2024 BLMS paper (Mineh, *Separability in Morse local-to-global groups*) still cites the RF ⟺ quasi-convex-subgroup-separability equivalence as conditional; a 2025 paper (Logan, *Algebraically hyperbolic groups*) states consequences conditionally on "if every hyperbolic group is residually finite". No solution claim exists in the literature as of this review (August 2026, web search).
+
+**Known positive classes** (all via virtual specialness or linearity): finite/virtually cyclic groups; virtually free groups; surface groups; closed hyperbolic 3-manifold groups (Agol, Wise); hyperbolic Coxeter groups; hyperbolic free-by-cyclic groups; one-relator groups with torsion; random (Gromov density-model) groups at density < 1/6 (Ollivier–Wise cubulation). **Frontier:** random groups at density ≥ 1/6, and in general any non-cubulated hyperbolic group.
+
+## Work done
+
+1. Confirmed the dataset wording character-for-character against the live author PDF (no correction needed; the truncation artifacts in note (b) of the source PDF's text layer do not affect Q 1.15 itself).
+2. Verified the two load-bearing citations (Kapovich–Wise 2000, Agol–Groves–Manning 2009) directly through the Crossref API — notably, a first guessed DOI for Kapovich–Wise (…/jabr.1999.8035) resolved to an unrelated Oberst paper and was discarded; the correct DOI is 10.1006/jabr.1999.8104. Two further citations (Haglund–Wise, Sela) verified indirectly via Crossref-deposited reference metadata.
+3. Surveyed current literature for any solution claim; none exists.
+
+**Mathematical analysis (why a quick resolution is blocked).** Residual finiteness asks that for each 1 ≠ g ∈ G there is a finite quotient separating g from 1. The Kapovich–Wise reduction shows the problem has a sharp dichotomy: either every hyperbolic group is residually finite, or there exists a non-elementary hyperbolic group with *trivial* finite residual — i.e., all obstructions concentrate in the worst possible case; there is no intermediate "separate some elements but not others" scenario at the level of the whole class. On the other side, Agol–Groves–Manning show a positive answer bootstraps itself via Dehn filling to full quasi-convex subgroup separability, which by Wise's hierarchy theorem forces virtual specialness of every cubulated hyperbolic group. Since the only known general route to residual finiteness is Malcev's theorem via linearity, and non-linear hyperbolic groups provably exist (source note (c); quaternionic-hyperbolic lattice quotients), any positive proof must construct finite quotients *without* linear representations — no such technique is known. A negative proof would require certifying that *all* finite quotients of some hyperbolic group are trivial, equivalently an infinite finitely presented torsion(-like) quotient of a hyperbolic group with no finite images; known constructions of monsters (Olshanskii) are not finitely presented/hyperbolic in the required way. Both directions are stuck on genuinely missing technology.
+
+## Result
+
+OPEN-TRIAGE. The problem is unsolved as of August 2026; the transcription is correct; the precise equivalence structure (Kapovich–Wise; Olshanskii) and the strongest conditional consequences (Agol–Groves–Manning + Wise) are documented and verified. No independent progress toward a solution is claimed — this is a famous problem where a solution attempt is beyond the scope of a bounded review.
+
+## What remains
+
+- Resolve the dichotomy: either prove all hyperbolic groups are residually finite (which by Agol–Groves–Manning + Wise would prove every cubulated hyperbolic group virtually special), or construct a non-elementary hyperbolic group with no nontrivial finite quotients (Gromov's suggested outcome).
+- Key sub-questions: (i) does every hyperbolic group act properly cocompactly on a CAT(0) cube complex (cubulation)? (ii) residual finiteness for random groups at densities ≥ 1/6; (iii) decide residual finiteness for explicit candidate classes, e.g. quotients of quaternionic hyperbolic lattices (the known non-linear examples).

research/AMR-010-0116.md

@@ -0,0 +1,159 @@
+---
+id: AMR-010-0116
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0116 — Rank of direct powers of a hyperbolic group (Dani Wise's conjecture)
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is accurate. The original wording, from M. Bestvina,
+*Questions in Geometric Group Theory* (updated July 2004), Question 1.16
+([author PDF](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), verified
+against the fetched source):
+
+> (Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of the group $G$, and let
+> $\operatorname{rank}(G^n)$ be the smallest number of generators of $G^n$.
+> **Conjecture.** If $G$ is word-hyperbolic then
+> $\displaystyle\lim_{n\to\infty}\operatorname{rank}(G^n)=\infty$.
+
+The notes in the source add: the conjecture is true for finite nontrivial $G$ by a pigeon-hole
+argument; it is true whenever it holds for some quotient of $G$ (in particular when $G$ has a
+proper finite-index subgroup); and it fails if there is an epimorphism $G \twoheadrightarrow G\times G$,
+with Wise having an example of a 2-generator *infinitely presented* $C'(1/6)$ small cancellation
+group witnessing this. The companion question Q 1.17 asks for "nice" (CAT(0), automatic, ...)
+groups where the conjecture fails.
+
+The question originates in Wise's own paper: D. T. Wise, *The rank of a direct power of a
+small-cancellation group*, Geom. Dedicata **94** (2002), 215–223,
+[doi:10.1023/A:1020968914280](https://link.springer.com/article/10.1023/A:1020968914280)
+(existence and abstract verified via Springer and the Weizmann Institute publication record).
+
+## Status / Literature
+
+**Open**, as far as I could verify (August 2026). I found no paper resolving the conjecture in
+either direction; recent surveys and papers that cite Wise's conjecture (e.g. Coulon–Fournier-Facio
+2023, below) still treat growth sequences of infinite groups as "mysterious". The following
+literature is verified (each item was checked against the arXiv API, the publisher page, or the
+reference list of a verified paper):
+
+- **D. T. Wise (2002)**, *The rank of a direct power of a small-cancellation group*, Geom. Dedicata
+  94, 215–223. Verified abstract: he constructs (i) a finitely generated $C'(1/6)$ group
+  $G_\infty$ with $\operatorname{rank}(G_\infty^n)=2$ for **all** $n$ — so the conjecture fails badly
+  for infinitely presented small cancellation groups; (ii) for each fixed $n$ a *finitely presented*
+  $C'(1/6)$ group $G_n$ with $\operatorname{rank}(G_n^n)=2$; (iii) a finitely generated $C'(1/6)$
+  group $D$ with an epimorphism $D\twoheadrightarrow D\times D$; and (iv) for each $m$ a residually
+  finite $C'(1/6)$ group with no proper subgroups of index $\le m$. He explicitly conjectures the
+  positive statement for word-hyperbolic groups.
+- **J. Wiegold & J. S. Wilson (1978)**, *Growth sequences of finitely generated groups*,
+  Arch. Math. (Basel) 30, 337–343 (MR 503347; verified as cited in Coulon–Fournier-Facio's
+  reference list). They prove: for a finitely generated **infinite simple** group $\Gamma$,
+  $d(\Gamma^p)\le d(\Gamma)+1$ for all $p\ge 1$ — infinite simple groups have essentially bounded
+  growth sequences, and no f.g. infinite simple group with non-constant growth sequence is known
+  (Wiegold–Wilson call this "irreducibly difficult"; see also Wiegold, *Is the direct square of
+  every 2-generator simple group 2-generator?*, Publ. Math. Debrecen 35 (1988), 207–209).
+- **A. Yu. Olshanskii (1995)**, SQ-universality of hyperbolic groups, Mat. Sb. 186 (verified
+  indirectly: invoked as [Ol'95] in Coulon–Fournier-Facio for the SQ-universality of torsion-free
+  non-elementary hyperbolic groups).
+- **R. Coulon & F. Fournier-Facio (2023)**, *Infinite simple characteristic quotients*,
+  [arXiv:2312.11684](https://arxiv.org/abs/2312.11684) (verified via the arXiv API and the fetched
+  paper). Theorem 1.5/4.1: every torsion-free non-elementary hyperbolic group $\Gamma$ admits
+  infinite, simple, characteristic quotients $\Gamma/N$ — all **not finitely presentable** —
+  containing any prescribed countable group. Combined with Wiegold–Wilson, these simple quotients
+  $S$ satisfy $d(S^p)\le d(S)+1\le d(\Gamma)+1$ for all $p$.
+- The finite case (nontrivial finite $F$ has $d(F^n)\to\infty$) is classical growth-sequence
+  theory initiated by Wiegold; Bestvina's notes record the elementary pigeon-hole argument.
+
+## Work done
+
+I verified the source wording (exact match; no correction needed), established the above verified
+literature base, and carried out the following rigorous reductions and observations (elementary but,
+as far as I can tell, the correct state of knowledge):
+
+**Proposition (reduction of the conjecture).** Let $G$ be a finitely generated group. Then
+$\operatorname{rank}(G^n)\to\infty$ in each of the following cases:
+1. $b_1(G)=\operatorname{rank}_{\mathbb Z}(G_{ab})\ge 1$: then $(G^n)_{ab}=G_{ab}^n$ surjects
+   $\mathbb Z^{\,n\,b_1(G)}$, so $\operatorname{rank}(G^n)\ge n\,b_1(G)\to\infty$.
+2. $G$ has some nontrivial finite quotient $F$: then $\operatorname{rank}(G^n)\ge
+   \operatorname{rank}(F^n)\to\infty$ by the finite case. In particular this holds if $G$ has a
+   proper finite-index subgroup (take its core).
+
+*Proof.* (1) is immediate since $\operatorname{rank}$ does not increase under quotients and
+$\operatorname{rank}(\mathbb Z^{m})=m$. For (2), $\operatorname{rank}(G^n)\ge
+\operatorname{rank}(Q^n)$ for every quotient $Q$ of $G$. $\square$
+
+**Corollary 1.** A counterexample to Wise's conjecture must be an infinite word-hyperbolic group
+that is *perfect* ($b_1=0$) and has *no nontrivial finite quotients at all*. In particular it must
+fail to be residually finite: since residual finiteness of an infinite group produces arbitrarily
+large finite quotients, **a positive answer to Bestvina's Q 1.15 (every hyperbolic group is
+residually finite — itself famously open) would imply Wise's conjecture.** So Q 1.16 is a strict
+weakening of Q 1.15, and any counterexample to Q 1.16 is also a counterexample to Q 1.15.
+
+**Corollary 2.** If $G\twoheadrightarrow G\times G$, then iterating gives
+$G\twoheadrightarrow G^{2^k}$, so $\operatorname{rank}(G^{2^k})\le\operatorname{rank}(G)$ and the
+conjecture fails for $G$. Wise's 2002 examples show such epimorphisms exist in the $C'(1/6)$ class
+when finite presentability is dropped. No hyperbolic (finitely presented) group $G$ with an
+epimorphism onto $G\times G$ is known, and no obvious invariant rules one out.
+
+**Observation 3 (the simple-group route is closed inside hyperbolic groups).** By Olshanskii's
+SQ-universality, every non-elementary hyperbolic group has every countable group embedded in some
+quotient. A *simple* group $S$ has only the quotients $S$ and $1$; if $S$ were a non-elementary
+hyperbolic group, every countable group would embed in $S$ itself — impossible, since the finitely
+generated $S$ has only countably many finitely generated subgroups while there are uncountably many
+isomorphism classes of finitely generated groups. Hence **no infinite simple hyperbolic group
+exists**, and the Wiegold–Wilson mechanism ($d(\Gamma^p)\le d(\Gamma)+1$ for infinite simple
+$\Gamma$) cannot produce a hyperbolic counterexample. Conversely, Coulon–Fournier-Facio show that
+every torsion-free non-elementary hyperbolic group *does* have (non-finitely-presentable) infinite
+simple quotients, whose growth sequences are bounded by Wiegold–Wilson. So quotient-based lower
+bounds on $\operatorname{rank}(G^n)$ coming from simple quotients cannot prove the conjecture; only
+finite quotients or the abelianization can, and any proof must use finite presentability of $G$
+essentially (hyperbolicity of quotients alone is insufficient, since non-finitely-presentable
+$C'(1/6)$ counterexamples exist).
+
+**Observation 4 (homological lower bounds fail).** The only Betti number giving a usable bound is
+$b_1$: $\operatorname{rank}(H)\ge b_1(H)$, and by Künneth $b_1(G^n)=n\,b_1(G)$ — this is exactly
+case (1). Higher homology gives nothing: there is no inequality $\operatorname{rank}(H)\ge
+b_2(H)-b_1(H)$ (e.g. $H=\mathbb Z\wr\mathbb Z$ is 2-generated with $H_2(H)$ free abelian of
+infinite rank, by the standard exterior-square computation of $H_2$ of a wreath product), and the
+naive Euler-characteristic bound "$\chi(H)\ge 1-\operatorname{rank}(H)$" fails already for
+$H=F_4^3$ ($\chi=(-3)^3=-27$ but $\operatorname{rank}=12<28=1-\chi$), since it requires
+cohomological dimension $\le 2$. Likewise $L^2$-Betti numbers of $G^n$ vanish for infinite $G$
+(Cheeger–Gromov), so $L^2$ methods give no rank bound.
+
+## Result
+
+Wise's conjecture (Bestvina Q 1.16) remains **open**. I did not solve it. What is established here:
+
+- Verified the original statement and the absence of a published solution; the only published
+  partial results are Wise's 2002 small-cancellation counterexamples outside the finitely
+  presented/hyperbolic world, and the classical finite-group growth-sequence theory.
+- Sharp reduction (Proposition + Corollary 1): the conjecture holds unless $G$ is infinite,
+  perfect, and has no nontrivial finite quotients; hence it is implied by residual finiteness of
+  hyperbolic groups (Q 1.15), and a counterexample would simultaneously refute Q 1.15.
+- Structural observations: no infinite simple hyperbolic group exists (Observation 3), so the known
+  bounded-growth mechanism for infinite simple groups cannot realize a hyperbolic counterexample;
+  but every torsion-free non-elementary hyperbolic group has non-finitely-presentable simple
+  quotients with bounded growth sequences (Coulon–Fournier-Facio + Wiegold–Wilson), so any proof
+  must exploit finite presentability in an essential way. Homological/$L^2$ invariants cannot
+  detect rank growth beyond $b_1$ (Observation 4).
+
+## What remains
+
+- Decide the conjecture in the residual case: $G$ infinite hyperbolic, $b_1(G)=0$, with no
+  nontrivial finite quotients. This is entangled with the residual finiteness problem (Q 1.15):
+  proving all hyperbolic groups residually finite settles Wise's conjecture affirmatively;
+  constructing a hyperbolic group with an epimorphism $G\twoheadrightarrow G^2$ (or even with
+  $\operatorname{rank}(G^n)$ bounded) would refute both.
+- Already the case $n=2$ is open in general: is $\operatorname{rank}(G^2)>
+  \operatorname{rank}(G)$ (or even $\ge \operatorname{rank}(G)+1$) for every non-elementary
+  hyperbolic $G$ with $b_1(G)=0$?
+- No growth *rate* is known in the cases where the conjecture holds only via finite quotients:
+  lower bounds on $\operatorname{rank}(G^n)$ in terms of the finite-quotient growth of $G$ would be
+  quantitative strengthenings (for finite simple $S$ one knows
+  $\operatorname{rank}(S^n)=\Theta(\log n)$-type behavior from Wiegold's theory).
+- Honesty note: the non-existence of a published solution is asserted on the basis of targeted
+  searches (arXiv API, web) rather than exhaustive review; if a resolution appeared very recently
+  or in an obscure venue, I did not find it. The citation [Bri22] in Coulon–Fournier-Facio's
+  introduction (approaches "of a different flavor" to growth sequences of infinite groups) was not
+  independently identified or verified.

research/AMR-010-0117.md

@@ -0,0 +1,62 @@
+---
+id: AMR-010-0117
+classification: OPEN-TRIAGE
+wording_corrected: yes
+---
+
+# AMR-010-0117 — Wise: nice groups whose direct powers have bounded rank
+
+## Problem (corrected statement if needed)
+
+The worklist transcription is a faithful rendering of the source, but it must be read together with the preceding item, Bestvina's Q 1.16, which defines the context. From [Bestvina, *Questions in Geometric Group Theory* (updated July 2004)](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), Section 1.4:
+
+> **Q 1.16 (Dani Wise).** Let $G^n$ denote the Cartesian product of $n$ copies of the group $G$, and let $\operatorname{rank}(G^n)$ be the smallest number of generators of $G^n$. **Conjecture.** If $G$ is word-hyperbolic then $\lim_{n\to\infty}\operatorname{rank}(G^n)=\infty$.
+>
+> **Q 1.17 (Dani Wise).** Find "nice" (e.g. CAT(0), automatic, ...) groups where this conjecture fails.
+
+So the problem is: **exhibit a "nice" infinite group $G$ — e.g. a CAT(0) group or an automatic group — such that $\operatorname{rank}(G^n)$ does not tend to infinity as $n\to\infty$** (equivalently, by monotonicity, such that $\operatorname{rank}(G^n)$ is bounded, hence eventually constant). The list notes that the conjectured divergence holds for nontrivial finite $G$, for any $G$ having a quotient for which it holds (in particular any $G$ with a proper finite-index subgroup), and that it fails whenever there is an epimorphism $G\twoheadrightarrow G\times G$; Wise had an example of a 2-generator *infinitely presented* $C'(1/6)$ small-cancellation group where it fails. Q 1.17 asks for such behavior in a tame class. (`C'(16)` in the source PDF is a rendering artifact for $C'(1/6)$.)
+
+## Status / Literature
+
+**Open.** No CAT(0), automatic, or otherwise "nice" group with $\operatorname{rank}(G^n)\not\to\infty$ is known as of this writing (August 2026), and I found no publication claiming one. What exists:
+
+- **Wise's counterexample without niceness.** D. T. Wise, [The rank of a direct power of a small-cancellation group](https://doi.org/10.1023/A:1020968914280), *Geom. Dedicata* **94** (2002), 215–223 (verified via Crossref; content confirmed via the [zbMATH review Zbl 1022.20013](https://zbmath.org/1864100)): he constructs an infinitely presented 2-generator $C'(1/6)$ group $G_\infty$ with an epimorphism $G_\infty\twoheadrightarrow G_\infty\times G_\infty$, hence $\operatorname{rank}(G_\infty^n)=2$ for all $n$; and for each *fixed* $n$ a finitely presented $C'(1/6)$ group $G_n$ with $\operatorname{rank}(G_n^n)=2$. Infinite presentation excludes CAT(0)/automatic (such groups are finitely presented); $G_n$ only controls one power, not the whole sequence.
+- **Finitely presented but not nice.** G. Baumslag and C. F. Miller III, [Some odd finitely presented groups](https://doi.org/10.1112/blms/20.3.239), *Bull. London Math. Soc.* **20**(3) (1988), 239–244 (verified via Crossref): a finitely presented group $G$ with a quotient isomorphic to $G\times G$. By Lemma 1 below this gives $\operatorname{rank}(G^n)\le\operatorname{rank}(G)$ for all $n$. Nothing in the literature asserts this group is CAT(0) or automatic; its construction (an amalgam built from the HNN-like group $\langle a,h,t\mid [a,h]=1,\,(a^2)^t=a^3,\,(h^2)^t=h^3\rangle$, criss-crossed along rank-2 free subgroups) gives no such structure.
+- **Finitely generated but not finitely presented.** D. Meier, [Non-Hopfian groups](https://doi.org/10.1112/jlms/s2-26.2.265), *J. London Math. Soc.* (2) **26** (1982), 265–270 (verified via Crossref), building on J. M. Tyrer Jones, *Direct products and the Hopf property*, *J. Austral. Math. Soc.* **17** (1974), 174–196: finitely generated groups $G\cong G\times G$, hence $\operatorname{rank}(G^n)$ constant.
+- **The finitely presented isomorphism problem is itself open.** Hirshon's question — does a nontrivial finitely presented group $G\cong G\times G$ exist? — remains open (Baumslag's problem lists; still listed as open in [Shpilrain's problem list](https://shpilrain.ccny.cuny.edu/gworld/problems/probFP.html) and discussed in [this 2015 exposition of Baumslag–Miller](https://berstein2015.wordpress.com/2015/04/18/a-group-with-a-quotient-isomorphic-to-the-direct-square/)). Q 1.17 is weaker in algebraic demand (bounded rank, not isomorphism) but stronger in geometric demand (CAT(0)/automatic).
+- **Growth-sequence theory (why examples are hard to find).** J. Wiegold and J. S. Wilson, Growth sequences of finitely generated groups, *Arch. Math. (Basel)* **30**(4) (1978), 337–343, and Wiegold's series on growth sequences of finite groups: for nontrivial finite $F$, $d(F^n)\to\infty$ (logarithmically for perfect $F$, linearly otherwise). Hence any $G$ with bounded $\operatorname{rank}(G^n)$ has no nontrivial finite quotients (see Lemma 3).
+- **Upper bounds for perfect groups.** M. R. Bridson, [Binary subgroups of direct products](https://ems.press/content/serial-article-files/44488), *Enseign. Math.* (2) **69** (2023), 399–416: if $G_1,\dots,G_m$ are perfect with $d(G_i)\le r$, then $d(G_1\times\cdots\times G_m)\le r\lfloor 1+\log_2 m\rfloor$; in particular $d(G^m)=O(\log m)$ for finitely generated perfect $G$. So the question is precisely whether the logarithmic rate can drop to bounded — this requires $G$ to have *no* finite quotients at all, where the growth-sequence machinery gives no lower bound.
+- **Candidate groups exist; the rank property is unknown for all of them.** Finitely presented CAT(0) groups with no nontrivial finite quotients do exist: Wise's thesis examples (fundamental groups of compact non-positively curved square complexes with no finite quotients; these groups are CAT(0), and by Niblo–Reeves, groups acting geometrically on CAT(0) cube complexes are biautomatic), and the Burger–Mozes finitely presented torsion-free simple lattices in products of trees. Whether any of these has $\operatorname{rank}(G^n)$ bounded is unknown. Even for infinite 2-generated simple groups, Wiegold's question whether $d(G^2)=2$ (Publ. Math. Debrecen 35 (1988), 207–209) is not settled in general (cited as motivation in Coulon–Fournier-Facio, [Infinite simple characteristic quotients](https://arxiv.org/pdf/2312.11684), arXiv:2312.11684).
+- The companion **Q 1.16** (hyperbolic groups) is also still open in general: it holds as soon as $G$ has a proper finite-index subgroup, so the only possible counterexamples are infinite hyperbolic groups with no proper finite-index subgroups — whose non-existence is entangled with the residual-finiteness conjecture for hyperbolic groups (Q 1.15).
+
+## Work done
+
+No computation was used; the following is pure reasoning, partly original assembly of standard facts into the constraints a solution must satisfy.
+
+**Lemma 0 (monotonicity).** $\operatorname{rank}(G^{n+1})\ge \operatorname{rank}(G^n)$, since $G^n$ is a quotient of $G^{n+1}$ by a coordinate projection. Hence $(\operatorname{rank}(G^n))$ either tends to $\infty$ or is eventually constant; "does not tend to infinity" $\iff$ bounded $\iff$ eventually constant.
+
+**Lemma 1 (the only known mechanism).** If there is an epimorphism $\varphi:G\twoheadrightarrow G\times G$, then $\operatorname{rank}(G^{2n})\le\operatorname{rank}(G^n)$ for all $n$, and consequently $\operatorname{rank}(G^m)\le\operatorname{rank}(G)$ for all $m$.
+*Proof.* $\varphi^n:G^n\twoheadrightarrow (G\times G)^n\cong G^{2n}$ is an epimorphism, giving the first inequality. Given $m$, choose $k$ with $m\le 2^k$; $G^m$ is a quotient of $G^{2^k}$, so $\operatorname{rank}(G^m)\le\operatorname{rank}(G^{2^k})\le\operatorname{rank}(G^{2^{k-1}})\le\cdots\le\operatorname{rank}(G)$. $\square$
+Every known group with bounded $\operatorname{rank}(G^n)$ (Tyrer Jones; Meier; Baumslag–Miller; Wise's $G_\infty$) comes from this mechanism or the stronger $G\cong G\times G$.
+
+**Lemma 2 (no solvable quotients).** If $\operatorname{rank}(G^n)$ is bounded then $G$ has no nontrivial solvable quotient; in particular $G$ is perfect.
+*Proof.* A nontrivial solvable group $S$ has nontrivial abelianization $S_{\mathrm{ab}}$ (if $S=[S,S]$ the derived series never terminates). For a finitely generated abelian group $A$, $d(A)=\max_p\dim_{\mathbb F_p}(A/pA)$, and this $p$-rank is additive over direct sums, so $d(A^n)=n\,d(A)$. Then $\operatorname{rank}(G^n)\ge d(S^n)\ge d((S_{\mathrm{ab}})^n)=n\,d(S_{\mathrm{ab}})\to\infty$. $\square$
+
+**Lemma 3 (no finite quotients).** If $\operatorname{rank}(G^n)$ is bounded then $G$ has no nontrivial finite quotient, i.e. $\widehat G=1$ (trivial profinite completion); in particular $G$ is not residually finite.
+*Proof.* Suppose $G\twoheadrightarrow F$ with $F$ finite nontrivial. Let $S$ be a simple quotient of $F$ (quotient by a maximal normal subgroup). Then $\operatorname{rank}(G^n)\ge d(S^n)$. If $S\cong \mathbb Z/p$, $d(S^n)=n$. If $S$ is nonabelian simple, $S^n$ has $n$ distinct maximal normal subgroups (coordinate kernels) with quotient $S$; a $d$-generated group has at most $|\operatorname{Epi}(F_d,S)|\le |S|^d$ epimorphisms onto $S$, and each such normal subgroup accounts for at least one (indeed $|\operatorname{Aut}(S)|$) of them, so $n\le |S|^d$, i.e. $d(S^n)\ge \log_{|S|} n$. Either way $d(S^n)\to\infty$. $\square$
+(This is the "simple pigeon-hole argument" alluded to in Bestvina's list, made explicit; the sharp logarithmic rate for finite perfect groups is Wiegold's theorem.)
+
+**Lemma 4 (non-Hopficity).** An epimorphism $\varphi:G\twoheadrightarrow G\times G$ with $G\neq 1$ forces $G$ to be non-Hopfian: $\pi_1\circ\varphi:G\twoheadrightarrow G$ has kernel $\varphi^{-1}(1\times G)\supsetneq\ker\varphi$, since the quotient is $\cong G\neq 1$. Hence (Mal'cev) a finitely generated residually finite group can never surject onto its own square — consistent with Lemma 3.
+
+**Corollary (constraints on a solution of Q 1.17).** Any CAT(0) or automatic group $G$ answering Q 1.17 must be: finitely presented, infinite, perfect with no nontrivial solvable quotients, and profinitely trivial (no proper finite-index subgroups). All known constructions achieving bounded rank additionally pass through a non-Hopfian epimorphism $G\twoheadrightarrow G\times G$. Groups satisfying the necessary conditions exist in the required classes (Wise's CAT(0)/biautomatic groups with no finite quotients; Burger–Mozes simple CAT(0) groups), so the obstruction is not the existence of candidates but the total absence of a technique for bounding $\operatorname{rank}(G^n)$ without an epimorphism onto $G\times G$ — and no CAT(0) or automatic group is known to admit such an epimorphism. Note also that non-Hopficity alone is not an obstruction in these classes: Wise constructed a non-Hopfian automatic group (*J. Algebra* **180** (1996), 845–847), so Lemma 4 does not rule out an automatic solution.
+
+## Result
+
+**OPEN-TRIAGE.** The problem is open. The best partial results in the literature are: (i) Wise's 2-generator infinitely presented $C'(1/6)$ group $G_\infty$ with $\operatorname{rank}(G_\infty^n)=2$ for all $n$; (ii) Wise's finitely presented $C'(1/6)$ groups $G_n$ with the $n$-th power 2-generated for each fixed $n$; (iii) Baumslag–Miller's finitely presented group with quotient $\cong G\times G$, which by Lemma 1 has $\operatorname{rank}(G^n)$ bounded — but none of these groups is CAT(0) or automatic, and the stronger Hirshon problem (finitely presented $G\cong G\times G$) is itself unresolved. My own contribution is the rigorous derivation of the necessary conditions (Lemmas 0–4): a solution must be finitely presented, perfect with no solvable or finite quotients (profinitely trivial), and non-residually-finite; candidates with these properties exist among CAT(0)/biautomatic groups (Wise's non-residually-finite square-complex groups, Burger–Mozes simple lattices), but boundedness of $\operatorname{rank}(G^n)$ is unknown for every one of them.
+
+## What remains
+
+1. Construct a CAT(0) (or automatic/biautomatic) group $G$ with an epimorphism $G\twoheadrightarrow G\times G$, or prove none exists. No technique currently produces self-similar epimorphisms within non-positive-curvature classes; Wise's and Baumslag–Miller's constructions use HNN/amalgam tricks with no CAT(0) control.
+2. Alternatively, bound $\operatorname{rank}(G^n)$ directly for a specific nice group with trivial profinite completion (e.g. a Burger–Mozes simple group) — even deciding whether $d(G^2)=d(G)$ for such $G$ is unknown, and Wiegold's 1988 question (is the direct square of every 2-generated simple group 2-generated?) remains open in general.
+3. Settle Q 1.16 (the hyperbolic case): equivalent, given Lemmas 2–3, to whether an infinite word-hyperbolic group can have no proper finite-index subgroup — a question subsumed by the residual finiteness conjecture for hyperbolic groups.
+4. Sharper quantitative question left by the literature: for finitely generated perfect $G$ with no finite quotients, is $\operatorname{rank}(G^n)$ necessarily $o(\log n)$, or can the Bridson/Wiegold $O(\log n)$ upper bound fail to be attained in either direction? Nothing seems to be known between "bounded" (open) and "$O(\log n)$".

research/AMR-010-0118.md

@@ -0,0 +1,58 @@
+---
+id: AMR-010-0118
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-010-0118 — Epstein's question: algorithmically computing the Čech cohomology of a hyperbolic group's boundary
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is accurate. Original wording (Question 1.18 of M. Bestvina, *Questions in Geometric Group Theory*, July 2004, verified against the author's copy at
+[www2.math.utah.edu/~bestvina/eprints/questions.pdf](https://www2.math.utah.edu/~bestvina/eprints/questions.pdf)):
+
+> **Q 1.18 (Epstein).** Let $G$ be a word-hyperbolic group and $\partial G$ its boundary. Is there an algorithm to compute $\check H^{i}(\partial G)\cong H^{i+1}(G,\mathbb{Z}G)$? In particular, is there an algorithm to decide whether $\check H^{i}(\partial G)\cong \check H^{i}(S^{2})$ for all $i$?
+
+The source adds two remarks: (a) if $\partial G$ has the cohomology of $S^2$ then it is homeomorphic to $S^2$ (Bestvina–Mess), and modulo a finite normal subgroup $G$ is then conjecturally commensurable to a hyperbolic 3-manifold group (Cannon's conjecture); (b) (Epstein–Sela) there *is* an algorithm to determine the number of ends of a hyperbolic group, i.e. the case of reduced $\check H^0$: compute $\delta$, build an automatic structure (detects finite/2-ended), and dovetail a search for a splitting over a finite subgroup (detects infinitely-ended).
+
+The isomorphism $\check H^{i}(\partial G;R)\cong H^{i+1}(G,RG)$ (as $G$-modules, any ring $R$) is the theorem of Bestvina–Mess, *The boundary of negatively curved groups*, J. Amer. Math. Soc. 4(3):469–481, 1991 (verified via multiple independent bibliographies, e.g. [arXiv:2110.13595](https://arxiv.org/pdf/2110.13595.pdf) and [arXiv:1302.3908](https://arxiv.org/pdf/1302.3908)).
+
+## Status / Literature
+
+**Open in general; solved for a substantial class; degree-0 case solved.** No publication found (searched through 2025) that gives a general algorithm, and none claiming undecidability either. Verified references:
+
+- **Bestvina–Mess 1991** (above): the duality $\check H^{i}(\partial G)\cong H^{i+1}(G,\mathbb{Z}G)$; $\partial G$ finite-dimensional; if $\partial G$ has the Čech cohomology of $S^n$ it is a homology manifold (and for $n=2$, homeomorphic to $S^2$).
+- **Bestvina, *Local homology properties of boundaries of groups*, Michigan Math. J. 43(1):123–139, 1996** (verified via [arXiv:1302.3908](https://arxiv.org/pdf/1302.3908) bibliography): $\partial G$ has the local homology of a homology manifold in the top degree when $H^*(G;\mathbb{Z}G)$ is concentrated appropriately.
+- **B. Barrett, *Computing JSJ decompositions of hyperbolic groups*, J. Topology 11(2):527–558, 2018** (verified via [arXiv:2210.09973](https://arxiv.org/pdf/2210.09973) bibliography): algorithmic JSJ, hence algorithmic detection of the Bowditch cut-point structure of $\partial G$ — topological information of Čech-type in degrees 0–1, but not the cohomology groups themselves.
+- **B. Barrett, PhD thesis *Detecting topological properties of boundaries of hyperbolic groups*, Cambridge, 2018** ([repository PDF](https://www.repository.cam.ac.uk/bitstreams/337a966c-b664-42cd-8678-75fe0aa3ea97/download)): explicitly frames Epstein's question (= Bestvina's Q 1.18) as **open in general**, and proves (Theorem 6.4.4): *there is an algorithm taking a presentation of a hyperbolic fundamental group $G$ of a graph of groups with free vertex groups and cyclic edge groups and returning presentations for the Čech cohomology $G$-modules of $\partial G$*; (Corollary 6.4.5): $H^*(G;\mathbb{Z}G)$ is computable for this class. Method: algorithmic JSJ + Otal decomposition spaces of line patterns in free groups.
+- **B. Barrett, *Computing the Čech cohomology of decomposition spaces*, [arXiv:1712.00780](https://arxiv.org/pdf/1712.00780), Dec. 2017** (arXiv listing verified; journal publication not verified — I cite only the preprint): the technical core of the thesis result; states that Epstein asked whether the Čech cohomology of $\partial G$ is computable as a $G$-module.
+- **V. Markovic, *Criterion for Cannon's conjecture*, GAFA 23(3):1035–1061, 2013, DOI 10.1007/s00039-013-0228-5** (verified via the [Springer PDF](https://link.springer.com/content/pdf/10.1007/s00039-013-0228-5.pdf) and the [Oxford GGT book bibliography](https://people.maths.ox.ac.uk/drutu/book.pdf)): Cannon's conjecture — and hence the *meaning* of a "yes" answer to the sphere-recognition part — remains open; Markovic proves it under an additional hypothesis of sufficiently many quasiconvex surface subgroups.
+- **Baumslag–Miller–Short, *Unsolvable problems about small cancellation and word hyperbolic groups*, Bull. LMS 26(1):97–101, 1994** (verified via [arXiv:2210.09973](https://arxiv.org/pdf/2210.09973) bibliography): hyperbolicity is a Markov property, hence undecidable from an arbitrary finite presentation; so any algorithm in this area must take "a presentation of a group promised to be hyperbolic" as input, not decide hyperbolicity itself.
+
+## Work done
+
+I did not solve the problem; I give a rigorous reduction showing exactly where the naive approach fails, which sharpens what a solution would have to provide.
+
+**Setup (effective semi-computation).** Let $G$ be torsion-free hyperbolic, given by a presentation (with the promise of hyperbolicity). By Papasoglu's detection algorithm one can effectively extract an explicit $\delta$ of $\delta$-hyperbolicity. Then the Rips complex $X=P_d(G)$ with $d\ge 4\delta+2$ is contractible, locally finite, and the $G$-action is free and cocompact, so
+$$\check H^{k}(\partial G)\;\cong\;H^{k+1}(G;\mathbb{Z}G)\;\cong\;H^{k+1}_c(X)\;=\;\varinjlim_{R}\,H^{k+1}(X,\,X\setminus B_R),$$
+where $B_R$ is the ball of radius $R$ about a basepoint. Each stage $H^{k+1}(X,X\setminus B_R)$ is the cohomology of an explicitly computable finite pair of simplicial complexes, and each bonding map is computable. So the *entire direct system is computable*.
+
+**The precise obstruction.** Since $G$ is of type FP (finite Rips $K(G,1)$), the limit $H^{k+1}(G;\mathbb{Z}G)$ is a finitely generated abelian group. Hence for each $k$ there exists $R_0(k)$ such that $H^{k+1}(X,X\setminus B_{R_0})$ already surjects onto the limit. However, nothing bounds $R_0(k)$ effectively: the kernels of the stage maps can keep collapsing at arbitrarily late radii, and computing the limit of a general computable direct system of finitely generated abelian groups with finitely generated limit is a $\Sigma_2/\Pi_2$-type task with no universal algorithm. The whole content of Epstein's question is therefore an **effective stability radius**: a computable function of (presentation, $\delta$, $k$) after which the system $\{H^{k+1}(X,X\setminus B_R)\}$ has stabilised. Equivalently (via the contracting-geodesics argument of Bestvina–Mess, which makes the complements $X\setminus B_R$ a model for the shape of $\partial G$): one needs effective control on the Čech expansion of $\partial G$ coming from the hyperbolicity constant alone.
+
+**Why the sphere-detection special case is at least as hard as it looks.** Deciding "$\check H^{i}(\partial G)\cong\check H^{i}(S^2)$ for all $i$" would, by Bestvina–Mess, decide whether $\partial G\cong S^2$, i.e. identify exactly the class of groups to which Cannon's conjecture applies. Since hyperbolicity is Markov/undecidable in general, even this decision problem must be posed relative to the promise class; within it, no algorithm is known, and the Markovic criterion suggests the sphere case is controlled by surface-subgroup existence — itself only known to be semi-decidable in general (Kahn–Markovic gives surfaces for 3-manifold groups, not an algorithm from a presentation).
+
+**Consistency check with the known partial result.** Barrett's theorem fits this analysis exactly: the algorithmic JSJ decomposition plus the combinatorial structure of decomposition spaces of line patterns is precisely a mechanism that produces the missing effective stability bound for the class of hyperbolic groups splitting over cyclic subgroups with free vertex groups. The obstruction in general is the absence of an effective description of $\partial G$ from which a finite Čech expansion can be certified.
+
+## Result
+
+- The problem is **open in general** as of this writing (2026-08); I found no solution in the literature and Barrett's 2018 thesis explicitly records it as open.
+- **Partial solution (literature):** computable for hyperbolic fundamental groups of graphs of groups with free vertex groups and cyclic edge groups (Barrett, Thm 6.4.4 + Cor 6.4.5 of the thesis; arXiv:1712.00780), and the degree-0/ends case is decidable (Epstein–Sela, per the source remark).
+- **My contribution:** a rigorous reduction of the general question to an *effective stability bound* for the computable direct system $\{H^{k+1}(P_d(G),P_d(G)\setminus B_R)\}$, showing that each stage and map is algorithmically computable from (presentation, $\delta$) and that the unique missing ingredient is a computable stabilisation radius; plus the observation that the $S^2$-detection subproblem is equivalent to recognising Cannon-conjecture groups from presentations, explaining its resistance.
+
+## What remains
+
+1. A general effective bound $R_0(\text{presentation},\delta,k)$ for the Rips-complement direct system — this is equivalent to a full solution of the first part of the question.
+2. Extension of Barrett's JSJ/decomposition-space method beyond free-vertex/cyclic-edge graphs of groups (e.g. to rigid hyperbolic groups with arbitrary one-ended structure, or groups whose boundaries have local cut points of general type).
+3. The $S^2$-recognition special case; even the weaker question "is $\check H^2(\partial G)\ne 0$ decidable?" appears open.
+4. The torsion case: passing from $H^{k+1}_c(X)$ (which sees only $\check H^k(\partial G)$) to $H^{k+1}(G,\mathbb{Z}G)$ when $G$ has torsion needs care, since hyperbolic groups are not known to be virtually torsion-free (residual finiteness of hyperbolic groups is itself open); Bestvina–Mess's module-level isomorphism handles this, but an algorithm must too.
+5. Caveat on verification: I verified every cited item against at least one independent bibliography or publisher page, but the arXiv API and Crossref endpoints were unreachable from this environment; Barrett's arXiv:1712.00780 publication venue (if any) was not confirmed, and the Papasoglu hyperbolicity-detection algorithm is cited from standard knowledge, not re-verified online in this session.

research/AMR-010-0119.md

@@ -0,0 +1,57 @@
+---
+id: AMR-010-0119
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0119 — Mitra's question: does inclusion of hyperbolic groups extend to a Cannon–Thurston map of boundaries?
+
+## Problem (corrected statement if needed)
+
+The worklist transcription matches the source verbatim (verified against the original PDF at
+https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, Question 1.19, "Maps Between Boundaries", PDF page 5):
+
+> **Q 1.19 (M. Mitra).** Let $G$ be a word-hyperbolic group and $H$ a word-hyperbolic subgroup. Does the inclusion $H \to G$ extend to a continuous map between the boundaries $\partial H \to \partial G$?
+
+Such an extension, when it exists, is called a **Cannon–Thurston (CT) map**. No correction of the wording is needed.
+
+## Status / Literature
+
+**Answer: NO in general.** The question was answered negatively by Baker and Riley in 2013. All citations below were verified against Crossref, the arXiv API, or publisher pages during this work.
+
+Positive results (map exists):
+
+- **Quasi-convex (undistorted) $H$:** a quasi-isometric embedding of hyperbolic spaces extends to a topological embedding of Gromov boundaries — classical (Gromov; see e.g. Kapovich–Benakli, *Boundaries of hyperbolic groups*, Contemp. Math. 296 (2002), DOI 10.1090/conm/296/05068, verified via Crossref reference data).
+- **Cannon–Thurston (1985/2007):** the original example — the fiber surface group of a closed hyperbolic 3-manifold fibering over $S^1$; the CT map is a group-equivariant Peano curve $S^1 \twoheadrightarrow S^2$. J. W. Cannon and W. P. Thurston, *Group invariant Peano curves*, Geom. Topol. 11 (2007), 1315–1355, DOI 10.2140/gt.2007.11.1315 (verified via Crossref reference data in 10.1017/fms.2013.4).
+- **Mitra 1998a:** CT maps exist when $H$ is an infinite normal subgroup of a hyperbolic $G$ (in particular for hyperbolic group extensions). M. Mitra, *Cannon–Thurston maps for hyperbolic group extensions*, Topology 37(3) (1998), 527–538, DOI 10.1016/S0040-9383(97)00036-0 (**verified via Crossref**). Point preimages are described by an ending-lamination theory: M. Mitra, *Ending laminations for hyperbolic group extensions*, GAFA 7(2) (1997), 379–402, DOI 10.1007/PL00001624 (verified via Crossref reference data).
+- **Mitra 1998b:** CT maps exist for vertex/edge groups in trees of hyperbolic spaces with quasi-isometric edge-to-vertex monomorphisms — this is the [Mit98b] cited in Bestvina's Q 1.19. M. Mitra, *Cannon–Thurston maps for trees of hyperbolic metric spaces*, J. Differential Geom. 48(1) (1998), 135–164, DOI 10.4310/jdg/1214460609 (**verified via Crossref**).
+- **Mj 2014:** CT maps exist for simply/doubly degenerate surface Kleinian groups (settling a question of Cannon–Thurston and Thurston's 1982 Problem 14); later for arbitrary finitely generated Kleinian groups (McMullen's conjecture; arXiv:1002.0996). M. Mj, *Cannon–Thurston maps for surface groups*, Ann. of Math. (2) 179(1) (2014), 1–80, DOI 10.4007/annals.2014.179.1.1 (**verified via the Annals of Mathematics journal page**).
+
+Negative result (the resolution):
+
+- **Baker–Riley 2013:** O. Baker and T. R. Riley, *Cannon–Thurston maps do not always exist*, Forum Math. Sigma 1 (2013), Paper No. e3, 11 pp., DOI 10.1017/fms.2013.4 (**verified via Crossref**; abstract: "We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries"), arXiv:1206.0505 (**verified via arXiv API**), MR3143716. This settles Q 1.19 in the negative.
+- **Matsuda–Oguni:** building on Baker–Riley, every non-elementary hyperbolic group embeds in *some* hyperbolic group with no CT map. Y. Matsuda and S. Oguni, *On Cannon–Thurston maps for relatively hyperbolic groups*, arXiv:1206.5868 (cited in the Baker–Riley paper; verified only as a cited arXiv preprint, not independently fetched).
+- **Distortion dichotomy:** subexponentially distorted subgroups of hyperbolic groups are quasi-convex (I. Kapovich, *The combination theorem and quasiconvexity*, Internat. J. Algebra Comput. 11(2) (2001), DOI 10.1142/S0218196701000553, verified via Crossref reference data), so CT maps exist there; Baker–Riley also showed CT maps can exist for extremely (non-recursively) distorted free subgroups of hyperbolic hydra: O. Baker and T. Riley, *Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra*, Groups Geom. Dyn. 14 (2020) (bibliographic details seen in reference lists of later papers; arXiv:1209.0815 cited within the Baker–Riley paper itself; DOI not independently fetched).
+
+## Work done
+
+1. **Source identification and wording check.** Fetched the Bestvina "Questions in Geometric Group Theory" PDF (July 2004 update) and confirmed Q 1.19 appears exactly as transcribed, on PDF page 5 in §1.6 "Maps Between Boundaries". The worklist text is a faithful transcription; `wording_corrected: no`.
+2. **Citation verification.** Verified the Baker–Riley paper directly via Crossref (10.1017/fms.2013.4) and the arXiv API (arXiv:1206.0505 — note: my first two guesses at the arXiv identifier, 1206.1482 and 1206.5368, returned unrelated physics/CS papers, a useful reminder that unverified identifiers are worthless); verified both Mitra 1998 papers via Crossref; verified Mj's Annals paper via the journal page. Cannon–Thurston 2007, Mitra 1997 (GAFA), and Kapovich 2001 were verified as DOI-asserted references inside the Crossref records fetched.
+3. **Extracted and checked the Baker–Riley argument** from the full text (arXiv:1206.0505v4). The construction:
+   - Let $C, C_i$ (on $c_1, c_2$) and $D_j, D_{ij}$ (on $d_1, d_2$) be long Rips-type positive words (e.g. $C = c_1 c_2 c_1 c_2^2 c_1 c_2^3 \cdots c_1 c_2^r$). For $r$ large,
+     $$G = \langle a, b, c_1, c_2, d_1, d_2 \mid a^{-1}b^{-1}ab = C,\ b^{-1}c_i b = C_i,\ (ab)^{-1} d_j (ab) = D_j,\ c_i^{-1} d_j c_i = D_{ij} \rangle$$
+     satisfies $C'(1/6)$, hence is hyperbolic (in fact a CAT($-1$) variant exists — Remark 9 of the paper, using Wise's pentagon pieces).
+   - $H = \langle b, d_1, d_2 \rangle$ is free of rank 3: the presentation is a tower of HNN extensions ($F(d_1,d_2) \leadsto G_{cd} \leadsto G_{bcd} \leadsto G$) and Britton's lemma, together with $F(c_1,c_2) \cap F(d_1,d_2) = \{1\}$ in $G_{cd}$, rules out any relation among $b, d_1, d_2$.
+   - **No CT map.** Mitra's criterion (Lemma 6 of the paper): the CT map exists iff $M(N) \to \infty$, where $M(N)$ measures how far $G$-geodesics between endpoints of $H$-geodesics staying outside $B(N)$ in $X_H$ must stay from $e$ in $X_G$. The words $w_n = b^{-n} a^{-n} d_1 a^{n} b^{n}$ are strongly Dehn-reduced, hence (Lemma 7, Kapovich–Short, via Strebel's appendix to Ghys–de la Harpe) label *geodesics* in the Cayley graph of $G$ passing through $e$ — so their endpoint pair is at $G$-distance $0$ from the identity along the path. But the relations rewrite $a^n b^n$ as a positive word in $ab, c_1, c_2$, and thence $w_n = u^{-1} d_1 u$ with $u^{-1}d_1u$ a positive word in $d_1, d_2$: the endpoints lie in $H$, and the $H$-geodesic between them stays at distance $\geq n$ from $e$ in $X_H$. Hence $M(n) = 0$ for all $n$, $M(N) \not\to \infty$, and no continuous extension $\partial H \to \partial G$ exists.
+4. **Why the positive theorems do not save the question.** The example clarifies the boundary of Mitra's theorems: $H$ is *not* normal in $G$, and in the factorization $H \hookrightarrow G_{bcd} \hookrightarrow G$ the middle group $G_{bcd}$ is hyperbolic and an HNN extension, but the defining monomorphisms fail the quasi-isometric-embedding hypothesis of the trees-of-spaces theorem — Baker–Riley show (Remark 8) that in fact *both* inclusions $H \hookrightarrow G_{bcd}$ and $G_{bcd} \hookrightarrow G$ admit no CT map, so the q.i.-embedding hypothesis in Mitra's tree theorem is essential, not an artifact. Also, $H$ has infinite height in $G$, so the example is consistent with Swarup's finite-height quasiconvexity question (Q 1.8 on Bestvina's list) remaining open.
+
+## Result
+
+Mitra's question (Bestvina Q 1.19) is **resolved in the negative**: Baker and Riley (Forum Math. Sigma 1, 2013, e3; DOI 10.1017/fms.2013.4; arXiv:1206.0505) constructed an explicit $C'(1/6)$ small-cancellation hyperbolic group $G$ on six generators containing a rank-3 free subgroup $H = \langle b, d_1, d_2 \rangle$ for which no Cannon–Thurston map $\partial H \to \partial G$ exists, with an elementary, fully rigorous proof via Mitra's $M(N)$ criterion and Dehn-reduced geodesics. Hence the answer to the question as posed is **no**, and the problem is solved in the literature. This is a literature triage, not an independent solution by me; I verified the source wording, the resolving paper's existence and abstract against Crossref and the arXiv API, and reconstructed its proof from the full text.
+
+## What remains
+
+- **Characterization problem:** given hyperbolic $H \leq G$, decide when a CT map exists. Known sufficient conditions: quasi-convexity; normality (Mitra); tree-of-spaces with q.i. edge maps (Mitra); Kleinian groups (Mj). Baker–Riley shows distortion alone is not the criterion: their $H$ is at least doubly-exponentially distorted, while hyperbolic hydra contain even more distorted free subgroups *with* CT maps. Subexponential distortion forces quasi-convexity (Kapovich), so the remaining gap is **Kapovich's question: is there an exponentially distorted hyperbolic subgroup of a hyperbolic group with no CT map?** (explicitly left open in Baker–Riley).
+- **Structure of CT maps when they exist:** description of point preimages beyond the normal-extension/ending-lamination case (this is Bestvina Q 1.20, attributed to Swarup).
+- **Relatively hyperbolic/generalizations:** CT maps for relatively hyperbolic groups and their subgroups (Matsuda–Oguni arXiv:1206.5868; later work of Mj–Pal and others), for CAT(0) groups with isolated flats, and non-existence results in the hierarchically hyperbolic setting are active topics (recent literature through 2025–2026 still cites Baker–Riley as the foundational counterexample).
+- Related open items on the same Bestvina list touched by this example: Swarup's finite-height question (Q 1.8) and the point-preimage problem (Q 1.20) remain open as far as I could verify within the fetch budget.

research/AMR-010-0120.md

@@ -0,0 +1,152 @@
+---
+id: AMR-010-0120
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0120 — Finiteness of Cannon–Thurston fibers for graphs of hyperbolic groups (Swarup)
+
+## Problem (corrected statement if needed)
+
+The transcription in `worklist/AMR-010-0120.md` matches the original source, Bestvina's
+"Questions in Geometric Group Theory", Question 1.20 (attributed to G. A. Swarup), so no
+correction was needed:
+
+> Suppose $G$ is a hyperbolic group which is a graph of hyperbolic groups such that all
+> edge-to-vertex inclusions are quasi-isometric embeddings. Mitra shows that each
+> vertex-group inclusion $V \hookrightarrow G$ induces a continuous Cannon–Thurston map
+> $\partial V \to \partial G$. Describe its point-preimages; in particular, show that the
+> map is finite-to-one.
+
+Equivalently (the form in which it is now standard): let $\Pi : X \to T$ be a tree of
+hyperbolic metric spaces satisfying the Bestvina–Feighn qi-embedded condition, with total
+space $X$ hyperbolic; for a vertex space $X_v$, Mitra proved the inclusion
+$X_v \hookrightarrow X$ admits a Cannon–Thurston map
+$\partial i : \partial X_v \to \partial X$; the question asks whether $\partial i$ is
+finite-to-one and for a description of its fibers (the "Cannon–Thurston lamination").
+
+## Status / Literature
+
+**Solved (second, quantitative part) as of March 2026**, subject to the caveat that the
+decisive paper is a very recent preprint that has not yet completed peer review.
+
+Verified citations (all checked against Crossref or the arXiv API):
+
+- Existence of the Cannon–Thurston map in this setting: M. Mitra,
+  "Cannon–Thurston maps for trees of hyperbolic metric spaces",
+  *J. Differential Geom.* 48 (1998), DOI `10.4310/jdg/1214460609` (verified via Crossref).
+- Related normal-subgroup case: M. Mitra, "Cannon–Thurston maps for hyperbolic group
+  extensions", *Topology* 37 (1998), 527–538, DOI `10.1016/s0040-9383(97)00036-0`
+  (verified via Crossref).
+- First affirmative partial answer to Swarup's question: I. Kapovich and M. Lustig,
+  "Cannon–Thurston fibers for iwip automorphisms of $F_N$",
+  *J. London Math. Soc.* 91 (2015), 203–224, DOI `10.1112/jlms/jdu069` (verified via
+  Crossref): for the free-by-cyclic group $F_N \rtimes_\phi \mathbb{Z}$ with fully
+  irreducible hyperbolic $\phi$, every fiber of the Cannon–Thurston map has cardinality
+  at most $2N$.
+- Generalization: S. Dowdall, I. Kapovich, S. J. Taylor,
+  "Cannon–Thurston maps for hyperbolic free group extensions",
+  *Israel J. Math.* 216 (2016), 753–797, DOI `10.1007/s11856-016-1426-2` (verified via
+  Crossref): uniform finiteness of fibers for hyperbolic extensions of $F_N$ by purely
+  atoroidal convex cocompact subgroups of $\mathrm{Out}(F_N)$, with fibers described via
+  algebraic (Cannon–Thurston) laminations. Their Theorem 6.3 explicitly states that it
+  answers Swarup's question (Q 1.20 on Bestvina's list) in that setting.
+- **Full answer**: I. Bhattacharyya, R. Halder, N. Lazarovich, M. Mj,
+  "Finiteness of Cannon–Thurston fibers", arXiv:2603.22428 (submitted 23 March 2026;
+  verified via the arXiv API — 14 pages, math.GT/math.GR). Theorem 3.13 and
+  Corollary 3.15: if $G$ is a hyperbolic group split as a finite graph of hyperbolic
+  groups with the qi-embedded condition and $H$ is a vertex group, then the
+  Cannon–Thurston map $\partial i : \partial H \to \partial G$ is **uniformly
+  finite-to-one**, with the bound depending only on the hyperbolicity, qi-embedding and
+  valence parameters. The same is proved for metric graph bundles (Mj–Sardar setting),
+  recovering the Kapovich–Lustig and Dowdall–Kapovich–Taylor results and a result of
+  Ghosh as special cases. The authors state explicitly that this "answers a question of
+  Swarup [Bestvina's list, Question 1.20]".
+- Description of point-preimages (first part of the question): fibers of $\partial i$ are
+  exactly the pairs/sets of boundary points joined by *contracting bi-infinite ladders*
+  flowing along a unique ray in the Bass–Serre tree $T$ (boundary-flow description; this
+  is Proposition 3.10 of arXiv:2603.22428, attributed to the monograph
+  M. Kapovich and P. Sardar, *Trees of Hyperbolic Spaces*, AMS Math. Surveys and
+  Monographs 282 (2024), Chapter 8 — existence of the volume and its Chapters 8–9
+  verified via Crossref DOIs `10.1090/surv/282/08`, `10.1090/surv/282/09` — building on
+  Mitra's earlier ladder description). In the free-group-extension special cases the
+  fibers are described concretely by algebraic laminations (Kapovich–Lustig 2015;
+  Dowdall–Kapovich–Taylor 2016; Mj–Rafi 2018).
+
+Note: the classical Cannon–Thurston theorem (fibers of
+$\partial \widetilde F \to \partial \widetilde M$ for a closed hyperbolic 3-manifold
+fibering over the circle, identified with endpoint-pairs of the stable/unstable
+laminations, hence uniformly finite-to-one) is the motivating special case; I did not
+independently verify its journal citation (Geom. Topol. 11 (2007)) via API, but it is
+entirely standard.
+
+## Work done
+
+1. Confirmed the source: Bestvina's problem list, Question 1.20 (Swarup). The worklist
+   transcription is faithful; no wording correction required.
+2. Traced the literature: the question circulated for ~20 years with affirmative answers
+   only in free-group-extension special cases (Kapovich–Lustig 2015, fibers $\le 2N$;
+   Dowdall–Kapovich–Taylor 2016, uniform bound via algebraic laminations and index
+   theory). Verified all of these against Crossref.
+3. Located and verified (via the arXiv API) the March 2026 preprint
+   arXiv:2603.22428 by Bhattacharyya–Halder–Lazarovich–Mj, which settles the question in
+   precisely Mitra's tree-of-hyperbolic-spaces setting. I read the full introduction and
+   the core argument (Sections 2–3 of the HTML version) and checked the logical
+   structure of the proof, which is short and self-contained modulo two black-box inputs
+   from Kapovich–Sardar (existence of contracting ladders behind multiple-value points,
+   and hyperbolicity of ray-preimages):
+   - For a finite set $A$ of boundary points in one fiber $\partial i^{-1}(\zeta)$, every
+     triple of points in $A$ admits a boundary flow along the ray $[u,\eta)$ in the tree
+     determined by $\zeta$, and the coarse barycenters of the flowed triples form a
+     uniform quasigeodesic section of that ray (their Lemmas 3.8, 3.11).
+   - All these barycenter-sections converge to the same point $\zeta \in \partial X$,
+     hence eventually lie within a uniform distance $R$ of each other inside a single
+     vertex space $X_n$ (their Lemma 3.4).
+   - A soft compactness/valence argument (their Proposition 2.4): in a bounded-valence
+     $\delta$-hyperbolic graph, a set of boundary points whose triple barycenters all
+     meet a fixed ball has uniformly bounded cardinality — rays to distinct points must
+     separate on a fixed sphere, whose size is bounded by $D^{R' + 10\delta}$.
+   - This bounds $|A|$ uniformly, proving uniform finiteness for a ray of spaces; a
+     reduction lemma (their Lemma 3.12, using uniqueness of the ray $\eta$ attached to a
+     multiple-value point) passes from rays to the general tree, giving the group
+     statement: the CT map $\partial H \to \partial G$ is uniformly finite-to-one.
+   The argument is correct as far as I can check without verifying the Kapovich–Sardar
+   inputs, and it deliberately avoids the lamination/index-theory machinery of the
+   earlier special-case proofs.
+
+## Result
+
+Swarup's Question 1.20 is answered affirmatively in the literature:
+
+- (Finite-to-one part) The Cannon–Thurston map $\partial V \to \partial G$ is
+  **uniformly finite-to-one** for any hyperbolic group split as a finite graph of
+  hyperbolic groups with qi-embedded edge inclusions: Bhattacharyya–Halder–Lazarovich–
+  Mj, arXiv:2603.22428 (2026), Theorem 3.13 / Corollary 3.15.
+- (Point-preimage part) Fibers are characterized as the sets of boundary points pairwise
+  joined by contracting bi-infinite ladders flowing along a unique end of the
+  Bass–Serre tree (Kapovich–Sardar 2024, Ch. 8; used as Proposition 3.10 in the
+  preprint); in free-group-extension cases they are described by explicit algebraic
+  Cannon–Thurston laminations (Kapovich–Lustig 2015; Dowdall–Kapovich–Taylor 2016).
+
+Caveat on classification: the full solution is a preprint (March 2026) that, at the time
+of writing, has not appeared in a refereed venue. If one insists on peer-reviewed
+literature only, the status would be "partially solved" (uniform finite-to-one known for
+free-group extensions since 2015–2016; the general graph-of-groups case open until the
+2026 preprint). I classified it SOLVED-IN-LITERATURE because the preprint is public,
+verified to exist, its proof is short and checkable in outline, and the authors include
+the originator of the existence theory (Mj = Mitra).
+
+## What remains
+
+- Formal peer review / publication of arXiv:2603.22428.
+- Explicit or optimal bounds on fiber cardinality in the general tree-of-spaces setting
+  (the proof gives an existence-type uniform bound depending on the parameters; in the
+  free-by-cyclic iwip case the sharp-looking bound $2N$ is known, but no such concrete
+  bound exists in general).
+- A fully explicit "lamination" description of point-preimages in the general
+  graph-of-hyperbolic-groups case, comparable in concreteness to the algebraic
+  laminations of Kapovich–Lustig / Dowdall–Kapovich–Taylor (currently only the
+  ladder/boundary-flow characterization is available in general).
+- Analogous fiber-finiteness questions in settings where Cannon–Thurston maps exist but
+  are not covered by arXiv:2603.22428 (e.g. relatively hyperbolic or more general
+  coarse-bundles contexts).

research/AMR-010-0121.md

@@ -0,0 +1,41 @@
+---
+id: AMR-010-0121
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0121 — Thurston's Virtual Fibering Question
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is verbatim-correct. In Bestvina's "Questions in Geometric Group Theory" (updated July 2004), Question 1.21 reads:
+
+> **Q 1.21. (Thurston)** Is every closed hyperbolic 3-manifold finitely covered by one that fibers over the circle?
+
+Confirmed by fetching the source PDF directly (https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, §1.7, PDF page 6). This is one of Thurston's celebrated questions from his 1982 problem list ("Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Bull. AMS 6 (1982), 357–381), where he asked whether every closed hyperbolic 3-manifold has a finite cover with positive first Betti number and, more strongly, one that fibers over S¹.
+
+## Status / Literature
+
+**Solved affirmatively in 2012–2013.** The affirmative answer is a consequence of Agol's proof of the Virtual Haken Conjecture, which completed a long program; all citations below were verified against Crossref and/or the arXiv API.
+
+- **Kahn–Markovic (2012)**: "Immersing almost geodesic surfaces in a closed hyperbolic three manifold", Ann. of Math. 175 (2012), 1127–1190. DOI 10.4007/annals.2012.175.3.4 (verified via Crossref). They proved the Surface Subgroup Conjecture: π₁ of every closed hyperbolic 3-manifold contains a quasi-Fuchsian surface subgroup, providing the essential surface needed to start the hierarchy/cubulation.
+- **Wise**: cubulation program — hyperbolic 3-manifold groups (among many others) admit quasiconvex hierarchies and hence act properly and cocompactly on CAT(0) cube complexes ("cubulated hyperbolic groups"). Published as the monograph "The Structure of Groups with a Quasiconvex Hierarchy", Annals of Mathematics Studies 209, Princeton University Press, 2021 (research announcement: ERA-MS 16 (2009), 44–55, DOI 10.3934/era.2009.16.44; book DOI 10.1515/9780691213507 — both verified via Crossref).
+- **Agol (2013)**: "The virtual Haken conjecture", Doc. Math. 18 (2013), 1045–1087. DOI 10.4171/DM/421 (verified via Crossref); arXiv:1204.2810 (verified via arXiv API). Agol proved that every cubulated hyperbolic group is *virtually special*, hence virtually embeds in a right-angled Artin group, is linear, and has separable quasiconvex subgroups (using Haglund–Wise special cube complexes). The abstract states explicitly that the work "resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question."
+
+The deduction of virtual fibering from virtual specialness runs through virtual Betti number: a virtually special hyperbolic 3-manifold virtually retracts onto its quasi-convex subgroups, so one can produce finite covers with arbitrarily large b₁; a closed irreducible 3-manifold with b₁ ≥ 1 has a non-fiber class dual to a non-separating surface, and work of Thurston (norm on H₂) together with the separability of the surface subgroups yields a finite cover in which some primitive class in H¹(·;ℤ) is represented by a fibration over S¹. (Precisely: Agol's theorem gives virtual RFRS — residual finiteness of the rings ℚ[Γ/Γᵢ] — and Agol's earlier criterion "Criteria for virtual fibering", J. Topol. 1 (2008), 269–284, then upgrades virtual positive b₁ to virtual fibering.)
+
+## Work done
+
+- Verified the exact wording of Q 1.21 against the source PDF (transcription matches verbatim; no correction needed).
+- Verified the existence and bibliographic data of the three load-bearing references via the arXiv API (Agol, arXiv:1204.2810) and Crossref (Agol DOI 10.4171/DM/421; Kahn–Markovic DOI 10.4007/annals.2012.175.3.4; Wise monograph DOI 10.1515/9780691213507 and announcement DOI 10.3934/era.2009.16.44).
+- Reconstructed the logical chain of the solution: Surface Subgroup Theorem (Kahn–Markovic) ⇒ cubulation of hyperbolic 3-manifold groups (Wise's hierarchy program) ⇒ virtually special (Agol, via Haglund–Wise + the Malnormal Special Quotient Theorem of Wise and its generalization in the Agol–Groves–Manning appendix) ⇒ virtual RFRS ⇒ virtual fibering (Agol's 2008 criterion).
+
+No independent mathematical contribution is possible or needed: the question was answered completely in the affirmative over a decade ago.
+
+## Result
+
+**Yes** — every closed hyperbolic 3-manifold has a finite-sheeted cover that fibers over the circle. This follows from Agol's theorem (Doc. Math. 18 (2013), 1045–1087) that cubulated hyperbolic groups are virtually special, combined with Wise's cubulation of hyperbolic 3-manifold groups and Agol's virtual-fibering criterion. The conjecture is fully resolved.
+
+## What remains
+
+Nothing for the stated question. Open descendants in the same circle of ideas include: quantitative control on the degree of the fibering cover (no effective bound is known in general); the analogous virtual questions in higher dimensions and for non-compact/finite-volume cases (already covered in dimension 3); and whether arbitrary hyperbolic groups (not necessarily 3-manifold groups) are virtually special or residually finite (cf. Bestvina Q 1.15, still open).

research/AMR-010-0123.md

@@ -0,0 +1,51 @@
+---
+id: AMR-010-0123
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0123 — Kan–Thurston theorem for CAT(-1) / word hyperbolic groups
+
+## Problem (corrected statement if needed)
+
+The dataset transcription was checked against the source, Bestvina's "Questions in Geometric Group Theory" (updated July 2004), Question 1.23, and matches it verbatim (attributed to Ian Leary):
+
+> Is there a version of the Kan–Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any finite simplicial complex X, there is a locally CAT(-1) polyhedral complex Y and a map Y → X that is surjective on fundamental groups and induces an isomorphism on homology for any local coefficients on X.)
+
+No correction needed. (Recall the classical theorem: Kan–Thurston, *Every connected space has the homology of a K(π,1)*, Topology 15 (1976), no. 3, 253–258, DOI 10.1016/0040-9383(76)90040-9 — verified via Crossref: correct authors, journal, volume, pages.)
+
+## Status / Literature
+
+**Open**, to the best of my verification (literature checked through August 2026 via arXiv API and web search). No paper claiming a CAT(-1) or word-hyperbolic Kan–Thurston theorem was found. What is known:
+
+- **CAT(0) version — solved.** Ian J. Leary, *A metric Kan–Thurston theorem*, J. Topol. 6 (2013), no. 1, 251–284; arXiv:1009.1540, DOI 10.1112/jtopol/jts035 (verified on the arXiv abstract page). For every simplicial complex X he builds a locally CAT(0) cubical complex T_X with a homology isomorphism t_X : T_X → X (with the extra structure of an involution making the quotient map a homotopy equivalence). This is exactly the "CAT(0) in place of CAT(-1)" analogue that Bestvina's note attributes to Leary. In the introduction Leary notes that his proof, and every proof of Kan–Thurston he knows, **uses direct products to increase dimension**, and that products are an obstruction to CAT(-1) — "any proof of a locally CAT(-1) Kan–Thurston theorem would have to involve a new idea."
+- **Torsion-allowed (proper-actions) analogue — solved, even at homotopy level.** T. Januszkiewicz and J. Świątkowski, *Simplicial nonpositive curvature*, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 1–85 (Numdam, PMIHES_2006__104__1_0; fetched and read). Their Theorem M (= Corollary 22.4): *any finite complex K is homotopy equivalent to the classifying space for proper G-bundles of a CAT(-1) (hence Gromov hyperbolic) group G*. This answered the companion Question 1.24 (homotopy types of R_d(G)/G; cf. Bestvina's Jan 2005 update "Any homotopy type occurs"), but **not** Q 1.23: the group G has torsion, so the quotient is B G = EG/G, not a K(G,1), and one cannot read off a torsion-free hyperbolic group realizing the homology of K.
+- **CAT(0) + duality-group refinement.** Raeyong Kim, PhD thesis (Ohio State, 2012, advisors Lafont and Leary; abstract and Ch. 2 fetched from OhioLINK): every finite complex has the homology of a CAT(0) cubical *duality* group, and every finite complex is homotopy equivalent to the classifying space for proper bundles of a virtual Poincaré duality group. Again CAT(0), not CAT(-1).
+- **2-dimensional case.** Bestvina's note under Q 1.23 states Leary can do the 2-dimensional cases using CAT(-1) or small-cancellation groups. I verified this statement exists in the list; I did not find a published paper containing the proof (it may be folklore/unpublished), so this partial result is second-hand.
+- **Classical refinements** (cited inside the verified sources above, not independently Crossref-checked): Baumslag–Dyer–Heller, *The topology of discrete groups*, J. Pure Appl. Algebra 16 (1980) — finite simplicial models; Hausmann (1979) — the realizing group can be taken to be a duality group.
+- Searches of the arXiv API (`all:"Kan-Thurston"`, and `"Kan-Thurston" AND hyperbolic`, sorted by date) and general web searches for 2013–2025 work turned up only citations of Leary's CAT(0) theorem, not a solution of the CAT(-1) question.
+
+## Work done
+
+I did not solve the problem; below is a rigorous analysis of the landscape and of why the standard techniques fail, which also identifies precisely what a solution would require.
+
+1. **The two formulations in the question are essentially one problem.** If Y is a finite locally CAT(-1) polyhedral complex, π1(Y) is word hyperbolic (cocompact proper action on the CAT(-1), hence δ-hyperbolic, universal cover, plus Švarc–Milnor). Conversely a torsion-free word-hyperbolic group G has a finite K(G,1) (Rips complex R_d(G) for large d is a finite model for EG, and with torsion-free G this is EG). So the question is equivalently: *is every finite simplicial complex X homology-equivalent (with arbitrary local coefficients, π1-surjectively) to BG for some torsion-free word-hyperbolic group G?*
+
+2. **Why Kan–Thurston-type proofs cannot be naively hyperbolized.** Every known proof (Kan–Thurston via acyclic groups and + -construction-like steps; Baumslag–Dyer–Heller; Hausmann; Leary's metric version) builds dimension by taking **products** of lower-dimensional acyclic pieces. A product of two non-positively curved spaces is CAT(0) but never CAT(-1): it contains isometrically embedded Euclidean planes, and its fundamental group contains Z², destroying hyperbolicity. Leary's building blocks are tesselated CAT(0) n-gons made of unit squares, and his inductive gluing functors L, M : S(X) → C(n) take products with these blocks at every stage. There is no known supply of *acyclic* (or suitably acyclic-with-local-coefficients) compact locally CAT(-1) complexes in arbitrary dimension that could play the same role; constructing one is already the heart of the problem.
+
+3. **Hyperbolization does not solve it.** Gromov/Charney–Davis-type strict hyperbolization produces, for any finite complex K, a locally CAT(-1) complex h(K) with a natural map h(K) → K, but that map is not a homology isomorphism with local coefficients: e.g. strict hyperbolization of S^n is a closed **aspherical** n-manifold mapping to S^n with degree ±1 — surjective, not injective, on H_n. Hyperbolization changes the homology; Kan–Thurston changes π1 while *preserving* homology. The two constructions are orthogonal.
+
+4. **The torsion obstruction in the one solved hyperbolic analogue.** Januszkiewicz–Świątkowski realize every finite K as EG/G with G a CAT(-1) group, via developments of simplices of *finite* groups (their Theorem H). Passing to a torsion-free subgroup Γ ≤ G of finite index makes EG/Γ a finite-sheeted cover — still a K(Γ,1) only if the Γ-action is free, which it is (Γ torsion-free, action proper), so EG/Γ is aspherical — but then EG/Γ has the homotopy type forced by the cover, **not** that of K. The covering trick destroys the prescribed homotopy/homology type. This is the exact point where their method cannot be promoted to answer Q 1.23.
+
+5. **No obstruction is known, so the answer is plausibly "yes".** There is no known homological restriction on torsion-free hyperbolic groups that would prevent Kan–Thurston realization: they are type F (hence FP over Z), can have arbitrarily large cohomological dimension (Januszkiewicz–Świątkowski's hyperbolic Coxeter groups), and arbitrary finitely generated homology in each degree is realizable by *some* finitely presented group, with hyperbolic examples (e.g. via Rips-type constructions applied to groups with prescribed homology) giving partial realization results. The 2-dimensional case (Leary, per Bestvina's note) is solved via small cancellation — and small cancellation is precisely a theory of 2-dimensional locally CAT(-1)-ish acyclic-ish complexes; the failure mode in higher dimensions is the absence of a higher-dimensional small-cancellation theory rich enough to produce the needed acyclic pieces (the Januszkiewicz–Świątkowski k-large/systolic theory is such a theory but, applied via complexes of finite groups, inherently produces torsion).
+
+## Result
+
+**OPEN-TRIAGE.** The problem as stated (CAT(-1) or word-hyperbolic Kan–Thurston) remains open as of August 2026. Solved neighbours: the CAT(0) version (Leary 2013, verified), the proper-actions/torsion-allowed CAT(-1) analogue at homotopy level (Januszkiewicz–Świątkowski 2006, Theorem M, verified), and reportedly the 2-dimensional case (Leary, stated in the source list, publication not located). I analyzed the standard approaches and identified two concrete barriers: (a) all Kan–Thurston proofs raise dimension via products, which create flats and hence only CAT(0); (b) the only known CAT(-1) realization theorem uses torsion in an essential way, and passing to torsion-free subgroups destroys the prescribed homology type.
+
+## What remains
+
+- Construct, in every dimension, compact aspherical locally CAT(-1) complexes that are acyclic (or acyclic relative to prescribed local coefficient systems) — the hyperbolic analogue of Leary's tesselated CAT(0) n-gons and Kim's CAT(0) acyclic duality-group blocks — together with gluing lemmas preserving CAT(-1) that can replace the product step in the Kan–Thurston induction.
+- Equivalently: find a hyperbolization procedure that preserves homology with arbitrary local coefficients (none known; strict hyperbolization provably does not).
+- Or prove impossibility: find a homological/finiteness obstruction distinguishing homology of (torsion-free) hyperbolic groups from homology of arbitrary finite complexes. No such obstruction is known.
+- Follow-up literature check worth doing: locate a published account of Leary's 2-dimensional CAT(-1) case (announced in Bestvina's list), and monitor for new work building on systolic/k-large techniques or on Ontaneda-style Riemannian hyperbolization.

research/AMR-010-0201.md

@@ -0,0 +1,187 @@
+---
+id: AMR-010-0201
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0201 — Swarup's question: Dehn twists and Out(G) for CAT(0) groups
+
+## Problem (corrected statement if needed)
+
+The worklist transcription matches the original source verbatim; no correction was
+needed. Source: M. Bestvina, *Questions in Geometric Group Theory*, Question 2.1
+(attributed to Swarup), [author-hosted PDF, updated July 2004](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)
+(fetched and checked directly; the updated version carries no status note on Q 2.1):
+
+> (Swarup) Is there a proof of Johannson's theorem that Out(π1M) is virtually
+> generated by Dehn twists for M a Haken 3-manifold along the lines of Rips–Sela's
+> theorem that Out(G) is virtually generated by Dehn twists for torsion-free
+> 1-ended hyperbolic G? Is this true for CAT(0) groups? In particular, if G is a
+> CAT(0) group and Out(G) is infinite, does G admit a Dehn twist of infinite order?
+
+Here a *Dehn twist* (definition given in the list itself) is an automorphism coming
+from a one-edge splitting: if G = A *_C B and t ∈ Z(C), the twist fixes A pointwise
+and conjugates B by t; similarly for HNN extensions.
+
+The question has three parts:
+1. Find a Rips–Sela-style proof of Johannson's theorem (Out(π1M) virtually generated
+   by Dehn twists for Haken 3-manifolds; Johannson, *Homotopy equivalences of
+   3-manifolds with boundaries*, LNM 761, Springer 1979).
+2. Does the Rips–Sela/Johannson picture hold for CAT(0) groups?
+3. (In particular) For a CAT(0) group G with Out(G) infinite, must G admit a Dehn
+   twist of infinite order?
+
+## Status / Literature
+
+**The question is resolved in the literature: the general CAT(0) form (parts 2 and 3)
+has a NEGATIVE answer, while the answer is positive for every natural "structured"
+subclass (hyperbolic, toral relatively hyperbolic, isolated-flats CAT(0) partially,
+and special/cubulated groups up to finite index). Part 1 is subsumed by the modern
+relatively-hyperbolic machinery.** Verified items:
+
+- **Hyperbolic groups (positive).** Rips–Sela: Out(G) virtually generated by Dehn
+  twists for torsion-free 1-ended hyperbolic G [E. Rips, Z. Sela, *Structure and
+  rigidity in hyperbolic groups I*, Geom. Funct. Anal. 4 (1994) — cited as [RS94] in
+  both Carette's and Fioravanti's papers below]. Strengthened by
+  [G. Levitt, *Automorphisms of hyperbolic groups and graphs of groups*, Geom.
+  Dedicata 114 (2005) 49–70](https://arxiv.org/abs/math/0212088) (verified via arXiv
+  API and by reading the arXiv text): his Theorem 1.4 states that for a one-ended
+  hyperbolic group G, Out(G) is infinite **iff** G splits over a virtually cyclic
+  subgroup with infinite centre — exactly the "infinite Out ⟹ infinite-order Dehn
+  twist" dichotomy (he notes Swarup suggested the problem). His Proposition 3.1 gives
+  a complete presentation of the group of twists of any graph of groups, which I used
+  below.
+- **CAT(0) with isolated flats (partial positive).**
+  [D. Groves, *Limits of (certain) CAT(0) groups, I: Compactification*, Algebr. Geom.
+  Topol. 5 (2005) 1325–1364](https://msp.org/agt/2005/5-4/agt-v5-n4-p03-p.pdf)
+  (fetched directly), Theorem 5.9: if Γ is torsion-free, acts properly and
+  cocompactly on a CAT(0) space with isolated flats, and flat stabilisers are
+  abelian, then Out(Γ) infinite ⟹ Γ splits over a finitely generated free abelian
+  group. Groves explicitly says this "partially answers a question of Swarup
+  (see [Bestvina, Q 2.1])".
+- **Coxeter groups (special case).**
+  [M. Carette, *Virtually splitting the map from Aut(G) to Out(G)*,
+  arXiv:1301.4446](https://arxiv.org/abs/1301.4446) (verified via arXiv API)
+  explicitly quotes Q 2.1 and discusses the Coxeter case; twist-rigid Coxeter groups
+  (Caprace–Przytycki) have finite Out.
+- **General CAT(0) groups (negative), and special groups (sharp positive).**
+  [E. Fioravanti, *Generators for automorphisms of special groups*,
+  arXiv:2601.22789 (Jan 2026, 79 pp., preprint — not yet refereed)](https://arxiv.org/abs/2601.22789)
+  (fetched and read): "Swarup asked whether Out(G) is virtually generated by Dehn
+  twists for every CAT(0) group G [Bes, Q2.1]"; Theorem C: every special group G
+  (Haglund–Wise) has a characteristic finite-index subgroup G₀ with Out(G₀) virtually
+  generated by Dehn twists; Proposition B: there *are* special (hence CAT(0)) groups
+  whose Out is not virtually generated by Dehn twists ("poison subgroups", a rank-2
+  abelian phenomenon); and, decisively, the discussion after Theorem C (Example 8.3):
+  for general CAT(0) groups there are groups G such that **every finite-index
+  subgroup G₀ ≤ G has infinite Out(G₀) and not a single (non-identity) Dehn twist** —
+  "In particular, the most general form of Swarup's question [Bes, Q2.1] has a
+  negative answer." The examples are extracted from:
+  - [G. Italiano, B. Martelli, M. Migliorini, *Hyperbolic 5-manifolds that fiber over
+    S¹*, Invent. Math. 231 (2023) 1–38](https://arxiv.org/abs/2105.14795) (verified
+    via arXiv API; the Invent. Math. reference appears verbatim in Fioravanti's
+    bibliography, surfaced via a web-search snippet of his PDF),
+  - [D. Groves, J. F. Manning, *Special IMM groups*, to appear in Bull. Lond. Math.
+    Soc.](https://arxiv.org/abs/2205.11290) (verified via arXiv API — this is
+    Fioravanti's [GM23]),
+  - [B. Martelli, *A 4-dimensional pseudo-Anosov homeomorphism*,
+    arXiv:2511.10530](https://arxiv.org/abs/2511.10530) (verified via arXiv API —
+    this is Fioravanti's [Mar25]; among its consequences: a compact locally CAT(0)
+    space whose π1 is non-hyperbolic and contains **no Z×Z**, answering Gromov's
+    Closing Flat problem).
+- **Part 1 (Johannson via Rips–Sela).** The Rips–Sela shortening/JSJ machinery has
+  since been developed for toral relatively hyperbolic groups (work of
+  Guirardel–Levitt, cited in Fioravanti's introduction as [GL15b] for the statement
+  "toral relatively hyperbolic groups behave similarly", i.e. Out virtually generated
+  by Dehn twists; and Groves' [Gro05] above). Fundamental groups of Haken
+  3-manifolds are relatively hyperbolic with abelian/Seifert peripheral structure, so
+  the Rips–Sela-style analysis of Out(π1M) now exists in this framework.
+  *Caveat:* I did not re-verify the Guirardel–Levitt papers themselves in this
+  session (their JSJ monograph, *JSJ decompositions of groups*, Astérisque 395, 2017,
+  is standard), and I am not aware of a paper explicitly titled "Johannson via
+  Rips–Sela"; the statement is subsumed by the relatively hyperbolic theory.
+
+## Work done
+
+No computation was used; this is a literature triage plus independent elementary
+reasoning.
+
+1. **Confirmed the source and wording.** Fetched Bestvina's updated problem list and
+   matched Q 2.1 word-for-word (the list even includes the definition of Dehn twist
+   that the dataset transcription omitted).
+2. **An elementary counterexample to part 3 (own analysis).** Let
+   G = Z² *_Z Z² = ⟨a,b,c,d | [a,b]=[c,d]=1, a=c⟩, the π1 of two flat tori glued
+   along a simple closed geodesic of equal length — a 2-dimensional (locally) CAT(0)
+   group by Reshetnyak's gluing theorem (these are essentially the Croke–Kleiner
+   examples). Then:
+   - *Out(G) is infinite.* The shears b ↦ aᵏb (fixing a,c,d) and d ↦ aˡd (fixing
+     a,b,c) — i.e. elements of the stabiliser of a primitive vector in GL(2,Z)
+     applied independently to the two vertex groups — give a Z×Z subgroup of Out(G):
+     an element of Inn(G) acts on each abelian vertex group either trivially or moves
+     the other factor off itself (normal forms in the amalgam), so Inn(G) meets this
+     shear subgroup trivially.
+   - *Every Dehn twist from this splitting is trivial in Out(G).* By Levitt's
+     Proposition 3.1 (read from the paper), the group of twists is the quotient of
+     ∏ Z_{G_v}(G_e) by vertex relations (centres of vertex groups) and edge relations
+     (centres of edge groups). Here both vertex groups are abelian, so the vertex
+     relations kill everything: the twist group is trivial. (Consistently, a twist by
+     t ∈ C conjugating the abelian factor B is the identity since t ∈ Z(B).)
+   So an elementary CAT(0) group with infinite Out and no nontrivial Dehn twist from
+   its natural splitting already exists — I did **not** fully verify the stronger
+   claim that this G admits no infinite-order Dehn twist from *arbitrary* splittings
+   (that requires ruling out exotic G-trees), which is why the published
+   Fioravanti/IMM/Martelli examples (no Dehn twists at all, even in every
+   finite-index subgroup) are needed for the definitive negative answer. I could not
+   locate an older reference stating this amalgam as an explicit counterexample to
+   Swarup's question; Carette (2013) still phrases the CAT(0) "iff" as a question,
+   so the folklore status of the easy example is unclear to me.
+3. **Mechanism behind the definitive counterexamples (sketch, from the sources).**
+   In the IMM/Martelli fibering constructions one has a (relatively) hyperbolic
+   mapping-torus group π1(M) = π1(F) ⋊_φ Z with F a compact aspherical locally
+   CAT(0) 4-manifold. The monodromy φ has infinite order in Out(π1F) — otherwise
+   π1(M) would virtually split as π1(F) × Z, incompatible with (relative)
+   hyperbolicity — so Out(π1F) is infinite; while π1(F) has no Z×Z (Martelli's
+   pseudo-Anosov monodromy, [Mar25]) or is arranged so that no splitting supporting
+   an infinite-order Dehn twist exists even after passing to finite index. Since any
+   infinite-order Dehn twist forces a splitting over an infinite subgroup with
+   infinite centraliser (in particular a Z²), such groups answer part 3 negatively.
+
+## Result
+
+- **Part 3 (and hence part 2) for general CAT(0) groups: NO.** There are CAT(0)
+  groups G with Out(G) infinite — indeed with every finite-index subgroup having
+  infinite Out — and not a single nontrivial Dehn twist. Published explicitly by
+  Fioravanti (arXiv:2601.22789, Example 8.3), built from the Italiano–Martelli–
+  Migliorini / Groves–Manning / Martelli fibering constructions (2023–2025). An
+  elementary 2-dimensional example (Z² *_Z Z²) shows the same phenomenon for twists
+  of the natural splitting (my analysis, based on Levitt's twist-group computation).
+- **Part 2 for restricted classes: YES.** Hyperbolic groups (Rips–Sela; Levitt's
+  Theorem 1.4 gives the sharp "Out infinite ⟺ infinite-order Dehn twist exists"
+  form); toral relatively hyperbolic groups (Guirardel–Levitt, per Fioravanti's
+  introduction); CAT(0) groups with isolated flats and abelian flat stabilisers
+  (Groves' Theorem 5.9, splitting conclusion); special (cocompactly cubulated
+  Haglund–Wise) groups up to a characteristic finite-index subgroup (Fioravanti,
+  Theorem C, with the failure inside special groups exactly characterised by
+  "poison subgroups", Theorem E).
+- **Part 1: effectively yes** — the Rips–Sela program now covers the class of groups
+  containing all Haken 3-manifold groups (toral/relatively hyperbolic JSJ theory),
+  so Johannson's theorem is recovered by Rips–Sela-style arguments, though no paper
+  with that explicit title seems to exist.
+- Classification: **SOLVED-IN-LITERATURE** (the question's hoped-for general CAT(0)
+  analogue is false; the precise boundary of validity is now mapped out). Note the
+  decisive reference for the negative answer is a January 2026 arXiv preprint, not
+  yet refereed.
+
+## What remains
+
+- Refereed publication of Fioravanti's preprint (arXiv:2601.22789) would put the
+  negative answer on firm published footing; the underlying manifold constructions
+  (IMM23, GM23, Mar25) are published or well-circulated.
+- Rips' related question on the structure of Out(G) for arbitrary cocompactly
+  cubulated groups (broader than special groups) remains open — Fioravanti's results
+  cover the special case.
+- For Coxeter groups, the general "Out(G) infinite ⟺ infinite-order Dehn twist"
+  question raised by Carette in 2013 was not fully resolved in the sources I checked.
+- It would be a small service to record the elementary Z² *_Z Z² counterexample
+  (with a complete proof that no splitting yields an infinite-order twist) in the
+  literature explicitly; I could not find it stated as such.

research/AMR-010-0202.md

@@ -0,0 +1,75 @@
+---
+id: AMR-010-0202
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0202 — Gromov's question: finite-dimensional K(G,1) ⇒ proper isometric action on a complete CAT(0) space?
+
+## Problem (corrected statement if needed)
+
+The dataset transcription was checked against the source PDF and is **verbatim correct**; no correction needed.
+
+**Question 2.2 of [Bestvina, "Questions in Geometric Group Theory" (updated July 2004)](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf):**
+
+> (Gromov) If $G$ admits a finite dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?
+
+Key features of the statement that matter for the literature triage:
+
+- The CAT(0) space is only required to be **complete** — it need not be *proper* (i.e., closed balls may be non-compact; infinite-dimensional Hilbert spaces are allowed).
+- The action is only required to be **properly discontinuous** — no cocompactness, and isometries need not be semisimple (parabolics are allowed).
+- The hypothesis (finite-dimensional $K(G,1)$) forces $G$ to be torsion-free, finitely presented, and of type FP; it is far weaker than being a "CAT(0) group" in the standard sense (proper *and cocompact* action on a CAT(0) space).
+
+## Status / Literature
+
+**Open as of August 2026**, to the best of my verification. No solution (positive or negative) appears in the literature I could verify. Supporting evidence and surrounding results (all citations verified against the source PDF, Crossref, the arXiv API, or publisher pages):
+
+1. **Source.** [Bestvina's problem list](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), §2 "CAT(0) groups", Q 2.2 (PDF p. 7). The 2004 update gives no solution or partial-result annotation, unlike many other items on the list.
+
+2. **The question is open even at a much coarser level.** Button, [*"Groups acting purely loxodromically on products of hyperbolic graphs"*](https://arxiv.org/abs/2009.10575) (arXiv:2009.10575, 2020; verified via arXiv API), states in the introduction: *"It is an open question whether every countable group acts properly on some CAT(0) space, whereas every countable group $G$ does act properly on some hyperbolic space."* Q 2.2 is the special case of this for groups with a finite-dimensional $K(G,1)$; since even the all-countable-groups version is open, so is Q 2.2.
+
+3. **Dimension-gap results do NOT answer Q 2.2 (important nuance).** There is a body of work showing that the *CAT(0) dimension* of a group can exceed its geometric dimension:
+   - Bridson, "Length functions, curvature and the dimension of discrete groups", *Math. Res. Lett.* 8 (2001), 557–567, DOI [10.4310/MRL.2001.v8.n4.a14](https://doi.org/10.4310/MRL.2001.v8.n4.a14) (verified via Crossref reference records).
+   - Crisp, ["On the CAT(0) dimension of 2-dimensional Bestvina–Brady groups"](https://doi.org/10.2140/agt.2002.2.921), *Algebr. Geom. Topol.* 2 (2002), 921–936, DOI 10.2140/agt.2002.2.921 (verified via Crossref): Bestvina–Brady groups $\Gamma_K$ of geometric dimension 2 that do not act properly on any 2-dimensional CAT(0) space, but act properly cocompactly on 3-dimensional ones.
+   - Brady–Crisp, ["Two-Dimensional Artin Groups with CAT(0) Dimension Three"](https://doi.org/10.1023/A:1020962804856), *Geom. Dedicata* 94 (2002), 185–214 (verified via Crossref).
+   - Tomiyoshi, ["Parabolic isometries of CAT(0) spaces and CAT(0) dimensions"](https://msp.org/agt/2004/4-2/agt-v4-n2-p09-s.pdf), *Algebr. Geom. Topol.* 4 (2004) (verified via the MSP page): groups of geometric dimension 2 that do not act properly on any proper CAT(0) space of dimension 2 by *semisimple* isometries — **but which do act properly on proper CAT(−1) spaces (of higher dimension) once parabolics are allowed** (his Theorem 1.1, 1.2, Corollary 5.1).
+   
+   All of these concern *proper* CAT(0) spaces of a *bounded dimension*, often with semisimplicity imposed. Q 2.2 allows arbitrary complete (possibly non-proper, infinite-dimensional) CAT(0) spaces and arbitrary isometries, so none of these examples obstructs Q 2.2 — indeed Tomiyoshi's groups *satisfy* its conclusion.
+
+4. **Even the hyperbolic special case is open.** Gromov's question whether every word-hyperbolic group acts properly and cocompactly on a CAT(0) (or CAT(−1)) space — the "Jugendtraum" — is a famous open problem; see Nica, ["Two applications of strong hyperbolicity"](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-59/issue-2/Two-applications-of-strong-hyperbolicity/10.1215/21562261-2019-0002.pdf), *Kyoto J. Math.* 59 (2019), which calls it "still wildly open" (verified via Project Euclid). Since every hyperbolic group has a finite K(G,1) (Rips complex, mod finite subgroups — for torsion-free hyperbolic groups a finite K(G,1) exists), a positive answer to Q 2.2 in the hyperbolic case would already be a major advance; conversely a negative answer to Q 2.2 would most plausibly come from (or at least illuminate) this case.
+
+5. **The class of groups acting properly on complete CAT(0) spaces is very broad**, which makes a negative answer hard to engineer:
+   - It contains all CAT(0) groups, all a-(T)-menable (Haagerup) groups (Hilbert spaces are CAT(0)), and is closed under passing to subgroups and direct products.
+   - It even contains infinite finitely generated torsion groups: Schneeberger, ["Proper actions of Grigorchuk groups on a CAT(0) cube complex"](https://doi.org/10.1007/s10711-024-00948-6), *Geom. Dedicata* (2024), DOI 10.1007/s10711-024-00948-6 (verified via Springer).
+   - No algebraic or analytic consequence of "acts properly on some complete CAT(0) space" is known that some group with a finite K(G,1) could fail. Property (T) is **not** an obstruction (cocompact lattices in $\mathrm{Sp}(n,1)$, $n\ge 2$, have (T) yet act properly cocompactly on quaternionic hyperbolic space, which is CAT(−1)). Note the contrast with CAT(0) *cube complexes*: a proper cubical action implies the Haagerup property, so infinite property-(T) groups admit no proper cubical actions — but cube complexes are a much smaller class of CAT(0) spaces.
+   - Recent tool-building: Petyt, ["Hyperbolic models for CAT(0) spaces"](https://arxiv.org/abs/2207.14127) (arXiv:2207.14127; published in *Adv. Math.* 2024; verified via arXiv/Warwick repository) shows any group acting properly on a CAT(0) space inherits a well-behaved action on an associated hyperbolic space — but since every countable group acts properly on *some* hyperbolic space (see item 2), this yields no obstruction.
+
+6. Background monograph for all CAT(0) terminology: Bridson–Haefliger, *Metric Spaces of Non-Positive Curvature*, Springer 1999, DOI [10.1007/978-3-662-12494-9](https://doi.org/10.1007/978-3-662-12494-9) (verified via Crossref).
+
+## Work done
+
+- Verified the dataset wording character-for-character against the original Bestvina PDF (Q 2.2, p. 7): exact match, attribution "(Gromov)" included. `wording_corrected: no`.
+- Established that the frequently-cited "dimension gap" literature (Bridson 2001; Brady–Crisp 2002; Crisp 2002; Tomiyoshi 2004) answers only **stronger variants** (proper spaces, bounded dimension, semisimple isometries) and does not decide Q 2.2 as stated; on the contrary, Tomiyoshi's exotic examples *do* act properly on complete CAT(−1) spaces, so they confirm rather than refute the conjectural implication in those cases.
+- Reasoned through both directions:
+  - *Positive direction (attempted):* the naive strategy — equip the universal cover of a finite-dimensional $K(G,1)$ with a $G$-invariant CAT(0) metric — fails in general: there are closed aspherical manifolds admitting no non-positively curved metric, and finite aspherical complexes whose universal covers carry no NPC metric (this is precisely the content of the dimension-gap papers above). Allowing non-proper/infinite-dimensional CAT(0) spaces removes the dimensional obstruction in principle, but no general construction is known — indeed none is known even for arbitrary *countable* groups (Button's remark, item 2).
+  - *Negative direction (attempted):* any counterexample $G$ must be a group with finite K(G,1) that does not embed in any group acting properly on a complete CAT(0) space (the class is subgroup-closed). All standard candidates are excluded: it cannot be a subgroup of a CAT(0) group, a Haagerup group, a cubulated group, or a lattice in a rank-1 group. No known invariant (bounded cohomology, property (T), Dehn function, torsion) separates "finite K(G,1)" groups from this class. Note that Dehn-function obstructions (e.g., the Baumslag–Gersten group's enormous Dehn function) only obstruct *cocompact* actions on *proper* CAT(0) spaces (which force quadratic Dehn function via quasi-isometry to the space); a merely proper action carries no such isoperimetric constraint, since orbits are distorted.
+- Searched for post-2020 developments (arXiv API full-text queries on the question's exact phrasing and variants; web searches for 2021–2026 preprints). Found no claimed solution or partial resolution of Q 2.2 itself.
+
+## Result
+
+**Open — rigorous triage.** The problem is unsolved in both directions as of August 2026:
+
+- No group with a finite-dimensional $K(G,1)$ is known that provably fails to act properly discontinuously by isometries on a complete CAT(0) space.
+- No theorem establishes such an action for all (or even for all hyperbolic) groups with finite-dimensional $K(G,1)$.
+- The strongest surrounding facts: (i) the *more general* question for arbitrary countable groups is explicitly open (Button 2020); (ii) the *cocompact/proper-space/semisimple* strengthenings are known to fail even in geometric dimension 2 (Bridson, Brady–Crisp, Crisp, Tomiyoshi 2001–2004), but their counterexamples still satisfy the conclusion of Q 2.2; (iii) the hyperbolic special case (Gromov's Jugendtraum) remains open.
+
+I could not solve or make substantive new mathematical progress on the problem itself; the difficulty is that the hypothesis gives a *finite-dimensional, possibly non-positively-curved* classifying space while the conclusion allows *arbitrary* complete CAT(0) spaces, and the two sides are connected by no known construction or invariant.
+
+## What remains
+
+- The full question: construct, for every $G$ with finite-dimensional $K(G,1)$, a proper isometric action on a complete CAT(0) space — or produce a counterexample.
+- Natural attackable sub-problems:
+  1. **Hyperbolic case:** does every word-hyperbolic group act properly (not necessarily cocompactly) on some complete CAT(0) space? This is weaker than the open Jugendtraum and might be more accessible; note (T) hyperbolic groups would need non-cubical CAT(0) targets.
+  2. **Baumslag–Gersten-type examples:** groups with finite $K(G,1)$ and non-elementary-recursive Dehn functions are natural stress tests; no obstruction to proper CAT(0) actions is known for them, and no action is known either.
+  3. **Find any invariant** of discrete groups that is forced by proper actions on arbitrary complete CAT(0) spaces but is not already forced by proper actions on hyperbolic spaces (Petyt's work suggests such invariants may be scarce), or prove none exists — which would point to a positive answer.
+- Also open and strictly harder: the same question with "complete" strengthened to "proper", or with cocompactness added (false in general, by Tomiyoshi's Corollary 5.1 — those strengthenings are *known* to fail, unlike Q 2.2 itself).

research/AMR-010-0203.md

@@ -0,0 +1,152 @@
+---
+id: AMR-010-0203
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0203 — The Eilenberg–Ganea problem (cd 2 vs gd 3)
+
+## Problem (corrected statement if needed)
+
+The transcription matches the source verbatim; no correction needed. Bestvina's
+"Questions in Geometric Group Theory" (updated July 2004), Q 2.3 (repeated verbatim
+as Q 8.1), reads:
+
+> **(Eilenberg–Ganea)** Is there a group $G$ of cohomological dimension $2$ and
+> geometric dimension $3$?
+
+Precise form: writing $\operatorname{cd}(G)$ for the cohomological dimension of $G$
+over $\mathbb{Z}$ and $\operatorname{gd}(G)$ for the minimal dimension of a
+$K(G,1)$-complex, is it true that $\operatorname{cd}(G)=2$ implies
+$\operatorname{gd}(G)=2$? Equivalently: is every group of cohomological dimension 2
+the fundamental group of an aspherical 2-dimensional CW complex?
+This is the Eilenberg–Ganea conjecture (1957), the unique remaining case of the
+question whether $\operatorname{gd}(G) = \operatorname{cd}(G)$ for all groups $G$.
+
+## Status / Literature
+
+**Open** as of August 2026, and unchanged since 1957 in its original integral,
+torsion-free form. Verified literature (every item checked against Crossref / arXiv /
+publisher metadata):
+
+- S. Eilenberg, T. Ganea, *On the Lusternik–Schnirelmann category of abstract
+  groups*, Ann. of Math. 65 (1957), 517–518, DOI 10.2307/1970062 (verified via
+  Crossref). Origin of the conjecture; they prove
+  $\operatorname{cd}(G) \le \operatorname{gd}(G) \le \max(\operatorname{cd}(G),3)$.
+- J. R. Stallings, *On torsion-free groups with infinitely many ends*, Ann. of
+  Math. 88 (1968), 312–334, DOI 10.2307/1970577, and R. G. Swan, *Groups of
+  cohomological dimension one*, J. Algebra 12 (1969), 585–610, DOI
+  10.1016/0021-8693(69)90030-1 (both verified via Crossref). Together: $\operatorname{cd}(G)=1
+  \iff G$ is free nontrivial $\iff \operatorname{gd}(G)=1$. With the
+  Eilenberg–Ganea upper bound and standard resolution arguments one gets
+  $\operatorname{gd}(G)=\operatorname{cd}(G)$ whenever $\operatorname{cd}(G)\neq 2$;
+  **$\operatorname{cd}=2$ is the only possible exception**, and then
+  $\operatorname{gd}\in\{2,3\}$.
+- M. Bestvina, N. Brady, *Morse theory and finiteness properties of groups*,
+  Invent. Math. 129 (1997), 445–470, DOI 10.1007/s002220050168 (verified via
+  Crossref). The sharpest result on the problem: the Bestvina–Brady group $H_L$
+  associated to a flag triangulation of a finite acyclic non-simply-connected
+  2-complex $L$ has $\operatorname{cd}(H_L)=2$, and if
+  $\operatorname{gd}(H_L)=2$ then the Whitehead asphericity conjecture (Bestvina's
+  list, Q 2.4) fails. Hence **at least one of the Eilenberg–Ganea conjecture and the
+  Whitehead conjecture is false**; both remain open.
+- N. Brady, I. J. Leary, B. E. A. Nucinkis, *On algebraic and geometric dimensions
+  for groups with torsion*, J. London Math. Soc. (2) 64 (2001), 489–500, DOI
+  10.1112/s002461070100240x (verified via Crossref). The analogue for groups with
+  torsion **fails**: there are groups (certain Coxeter groups) whose algebraic
+  dimension for the family of finite subgroups is 2 while the minimal dimension of a
+  model for $E_{\mathcal{F}\mathrm{in}}G$ is 3. This shows the gap phenomenon is
+  real in the proper/Bredon setting, but says nothing about torsion-free groups.
+- M. Fluch, I. J. Leary, *An Eilenberg–Ganea phenomenon for actions with virtually
+  cyclic stabilisers*, Groups Geom. Dyn. 8 (2014), 135–142, DOI 10.4171/GGD/219
+  (verified via the EMS Press page): Bredon $\operatorname{cd}=2$ vs Bredon
+  $\operatorname{gd}=3$ for the family of virtually cyclic subgroups.
+- L. J. Sánchez Saldaña, *Groups acting on trees and the Eilenberg–Ganea problem for
+  families*, arXiv:1911.03457 (accepted version for Proc. AMS; verified via arXiv
+  API): more examples with the 2-vs-3 gap for the families of finite, bounded-rank
+  virtually abelian, and virtually polycyclic subgroups.
+- E. Martínez-Pedroza, L. J. Sánchez Saldaña, *Bowditch taut spectrum and dimensions
+  of groups*, arXiv:2107.10643 (verified via arXiv API): among other things, if
+  there is one finitely presented counterexample to Eilenberg–Ganea, then there are
+  continuously many pairwise non-quasi-isometric one-ended counterexamples.
+- M. Grant, E. Meir, I. Patchkoria, *Equivariant dimensions of groups with
+  operators*, arXiv:1912.01692 (verified via arXiv API): equivariant
+  Eilenberg–Ganea and Stallings–Swan theorems; the same exceptional case
+  ($\operatorname{cd}=2$, $\operatorname{gd}=3$) persists equivariantly.
+
+An arXiv full-text search for "Eilenberg-Ganea" sorted by date (arXiv API, run
+2026-08-04) returns no paper claiming a solution of the original conjecture.
+
+## Work done
+
+This was a literature-triage and reasoning task; no computation was used (per
+constraints). I verified the source wording directly from Bestvina's author-hosted
+PDF (the transcription in `worklist/AMR-010-0203.md` is exact, including the
+attribution "(Eilenberg-Ganea)"; the item also appears verbatim as Q 8.1 of the same
+list). I then verified each citation above against Crossref or the arXiv API — two
+guessed DOIs initially 404'd (Swan: correct suffix is `90030-1`, not `90030-4`;
+Brady–Leary–Nucinkis: correct DOI is `10.1112/s002461070100240x`), and were
+corrected via Crossref bibliographic queries.
+
+On the mathematics, the complete classical reduction is short and worth recording:
+
+1. **Only $n=2$ is open.** $\operatorname{cd}(G)\le\operatorname{gd}(G)$ is
+   immediate (cellular chains of a $K(G,1)$ give a free resolution). Eilenberg–Ganea
+   prove $\operatorname{gd}(G)\le\operatorname{cd}(G)+1$, and the standard
+   "Eilenberg–Ganea theorem" upgrades this to equality when
+   $\operatorname{cd}(G)\ge 3$: starting from a projective resolution one builds a
+   $K(G,1)$ inductively, and in dimensions $\ge 4$ the obstructions to keeping the
+   complex low-dimensional vanish; Stallings–Swan settles $\operatorname{cd}=1$
+   (free groups, $K(G,1)$ a graph). So a counterexample must have
+   $\operatorname{cd}=2$ and $\operatorname{gd}=3$ exactly.
+2. **The Bestvina–Brady dichotomy.** For a finite acyclic 2-complex $L$ with
+   $\pi_1(L)\neq 1$ and flag triangulation, the kernel $H_L$ of the "send every
+   vertex to $1$" map from the right-angled Artin group on the 1-skeleton to
+   $\mathbb{Z}$ is finitely presented, of type $FP_2$, with
+   $\operatorname{cd}(H_L)=2$. Bestvina–Brady show that if $H_L$ had a
+   2-dimensional $K(H_L,1)$, the chain-level consequences would force the universal
+   cover $\widetilde L$ — an acyclic, hence aspherical-candidate, 2-complex — to
+   have a non-aspherical subcomplex, contradicting Whitehead's conjecture. So a
+   positive answer to Whitehead (Q 2.4) yields counterexamples to Eilenberg–Ganea,
+   and a negative answer to Eilenberg–Ganea does not come cheap: **the two oldest
+   open problems in this area cannot both be true.**
+3. **Where counterexamples cannot hide.** Any counterexample $G$ is necessarily
+   non-free, torsion-free of $\operatorname{cd}=2$, and not the fundamental group of
+   any aspherical 2-complex; in particular it cannot be a knot group, a one-relator
+   group, or a (torsion-free) hyperbolic/CAT(0) group with a 2-dimensional model.
+   Candidate constructions all run through acyclic 2-complexes (Bestvina–Brady) or
+   through refinements of efficiency/deficiency obstructions (the "Q-gap" criterion:
+   for finitely presented $G$ with $\operatorname{cd}(G)=2$, the conjecture is
+   equivalent to $\operatorname{gap}(G;\mathbb{Q})=0$, cf. the homology-roses
+   literature), and every candidate simultaneously threatens the Whitehead
+   conjecture — which is precisely why the problem has resisted 65+ years.
+
+No new partial result beyond this synthesis was obtained; a genuine advance would
+require either settling Whitehead's conjecture or a fundamentally new construction
+of $K(G,1)$'s, which is beyond a literature-triage budget.
+
+## Result
+
+The problem is **open**. The transcription is correct as given. The state of
+knowledge is: (i) $\operatorname{cd}=2$ is the unique dimension in which
+$\operatorname{gd}=\operatorname{cd}$ is unknown; (ii) Bestvina–Brady (1997) reduce
+the conjecture, in the presence of explicit candidate groups, to the Whitehead
+asphericity conjecture — one of the two must fail; (iii) the analogous statements
+for groups with torsion and for Bredon cohomology with various families are
+**false** (Brady–Leary–Nucinkis 2001; Fluch–Leary 2014; Sánchez Saldaña 2019), so
+the 2-vs-3 gap is a genuine phenomenon in every variant that allows torsion; the
+torsion-free integral case remains untouched.
+
+## What remains
+
+- The original problem: exhibit a torsion-free group $G$ with
+  $\operatorname{cd}(G)=2$ and no 2-dimensional $K(G,1)$, or prove none exists.
+- Equivalently (Bestvina–Brady): decide the Whitehead asphericity conjecture — a
+  positive answer hands over the counterexamples $H_L$ immediately.
+- The rational variant $\operatorname{cd}_{\mathbb{Q}}(G)=2 \Rightarrow
+  \operatorname{gd}(G)=2$ for torsion-free groups is also open and strictly weaker;
+  Martínez-Pedroza–Sánchez Saldaña produce groups with $\operatorname{cd}_{\mathbb
+  Q}=2$ but $\operatorname{cd}_{\mathbb Z}=3$, a nearby but distinct phenomenon.
+- Any solution would likely need new methods for constructing aspherical 2-complexes
+  (or for obstructing them) that distinguish the torsion-free world from the
+  proper-action world where counterexamples are known.

research/AMR-010-0205.md

@@ -0,0 +1,117 @@
+---
+id: AMR-010-0205
+classification: SOLVED-IN-LITERATURE
+wording_corrected: no
+---
+
+# AMR-010-0205 — Branched covers of S×S along the diagonal admit no smooth NPC metric
+
+## Problem (corrected statement if needed)
+
+The dataset transcription matches Bestvina's source list verbatim; no correction was needed.
+Original wording (Bestvina, *Questions in Geometric Group Theory*, updated July 2004, Q 2.5,
+<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>):
+
+> (Exercise in [BGS85, p. 2]) Take a closed surface S of genus ≥ 2. Let V = S × S and let
+> Σ ⊂ V denote the diagonal. Let Ṽ be a nontrivially ramified finite cover of V along Σ.
+> Then Ṽ has a natural piecewise hyperbolic CAT(0) metric. Show that Ṽ admits no
+> C²-smooth Riemannian metric with curvature K ≤ 0.
+
+Here [BGS85] = W. Ballmann, M. Gromov, V. Schroeder, *Manifolds of Nonpositive Curvature*,
+Progress in Mathematics 61, Birkhäuser, 1985 (DOI 10.1007/978-1-4684-9159-3), where this is
+the first exercise of the book.
+
+Setup and why the premises hold:
+
+- The diagonal Σ is totally geodesic in the product of hyperbolic metrics on S × S, of
+  codimension 2. Ramified covers of an NPC Riemannian manifold along a totally geodesic
+  codimension-2 submanifold carry a natural locally CAT(0) (here piecewise hyperbolic)
+  length metric — an observation of Gromov; see also R. Charney, M. Davis, "Singular metrics
+  of nonpositive curvature on branched covers of Riemannian manifolds", Amer. J. Math. 115(5)
+  (1993), 929–1009 (DOI 10.2307/2375063; verified via Crossref).
+- Nontrivially ramified finite covers exist: S × S ∖ Σ is the configuration space of two
+  ordered points on S, and a Mayer–Vietoris computation (the normal bundle of Σ is TΣ, of
+  Euler number 2 − 2g) shows the meridian of Σ is torsion of order dividing 2g − 2 in
+  H₁(S × S ∖ Σ; ℤ); since 2g − 2 is even, at least a double branched cover exists for every
+  g ≥ 2. (My own check of the BGS premise; I verified the torsion bound, not the exact order.)
+
+## Status / Literature
+
+**Solved.** The exercise was carried out by Stephan Stadler:
+
+- S. Stadler, "An obstruction to the smoothability of singular nonpositively curved metrics
+  on 4-manifolds by patterns of incompressible tori", Geom. Funct. Anal. 25(5) (2015),
+  1575–1587. DOI 10.1007/s00039-015-0341-8 (verified via Crossref); arXiv:1312.2198 (Dec 2013;
+  verified via the arXiv API). Also Chapter/result of his LMU dissertation *New obstructions
+  to smooth nonpositively curved metrics in dimension 4* (advisor B. Leeb, defended 16 July
+  2014; verified at <https://edoc.ub.uni-muenchen.de/19614/>).
+
+Stadler's Theorem 1 (stated as "Exercise 1 in [BGS85]"): *Let V be a closed 4-manifold which
+admits a non-trivial finite branched covering β: V → Σ × Σ over the product of a hyperbolic
+surface Σ with itself with branching locus the diagonal ΔΣ. Then V admits no smooth
+Riemannian metric of nonpositive sectional curvature.* Since "no smooth NPC metric" is
+stronger than "no C² NPC metric", this settles Q 2.5 completely. The paper's abstract states
+it is "answering affirmatively a question of Gromov" and the introduction says "The purpose
+of this note is to do this exercise."
+
+Prior related milestone (different examples, first of their kind): M. Davis, T. Januszkiewicz,
+J.-F. Lafont, "4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings", Duke
+Math. J. 161(1) (2012), 1–28 (DOI 10.1215/00127094-1507259; verified via Crossref) — smooth
+4-manifolds with isolated ℤ²'s whose invariant flats are "knotted at infinity", impossible in
+smooth Hadamard 4-manifolds. Stadler's approach is complementary: the branched covers have
+*plenty* of ℤ²'s, forcing an over-dense pattern of flat tori.
+
+## Work done
+
+1. Confirmed the dataset wording against Bestvina's PDF (fetched directly): Q 2.5 is
+   transcribed verbatim; the only artifacts are typographical (˜V, C2, ≤0).
+2. Verified the resolution and every citation above against Crossref/arXiv (DOIs and the
+   arXiv abstract page 1312.2198), and read the argument in the arXiv HTML version.
+3. Checked why "cheap" obstructions cannot do the exercise, as my own sanity analysis:
+   - Ṽ is aspherical (its universal cover with the pulled-back metric is CAT(0), hence
+     contractible), so one cannot argue via contractibility.
+   - For a k-fold branched cover, χ(Ṽ) = kχ(V) − (k−1)χ(Σ) = (2g−2)(k(2g−2) + k − 1) > 0.
+     This is consistent with the sign of χ for NPC 4-manifolds (the 4-dimensional Hopf sign
+     question has an affirmative answer), so Euler characteristic gives no obstruction.
+   - For a double cover, Hirzebruch's branched-cover signature formula gives
+     σ(Ṽ) = 2σ(V) − ½[Σ]² = g − 1 ≠ 0, but nonzero signature is also no obstruction to NPC
+     in general (compact complex-hyperbolic surfaces have σ ≠ 0 and K < 0).
+   - Hence the obstruction is genuinely geometric, not characteristic-class or
+     fundamental-group-theoretic — consistent with the problem being open from 1985 to 2013.
+4. Summary of Stadler's proof (from the arXiv version): The universal cover X of Ṽ with the
+   singular CAT(0) metric contains two rigid convex product subsets interacting badly:
+   (a) lifts of "product blocks" Σ⁺ × Σ̄⁻ disjoint from the diagonal, convex subsets
+   Y₁ × Y₂ preserved by a product F × F of free subgroups (product rigidity à la
+   Monod/Schroeder), and (b) a product Z × ℝ whose cross-section Z contains an ideal
+   triangle, whose three flats come from flat half-planes in c × c ⊂ Σ × Σ orthogonal to
+   the diagonal along a nonperiodic simple geodesic — such flats branch along the singular
+   locus π⁻¹(ΔΣ) and are shown (Lemma 4) to be pointed Hausdorff limits of Γ-periodic flats,
+   hence quasi-isometry invariant (via Kleiner and Lang–Schroeder). One defines a
+   "coarse intersection" relation between flats that is a quasi-isometry invariant and, in
+   smooth Hadamard manifolds, coincides with transverse point intersection. The configuration
+   (conditions (i)–(vii)) therefore transfers to any CAT(0) space with a geometric
+   π₁(Ṽ)-action; but in a *smooth* Hadamard manifold it forces either two flats to share a
+   quadrant and coincide, or Tits-distance < π between antipodal ideal points — a
+   contradiction (Claims 1–2). Hence π₁(Ṽ) acts geometrically on no Hadamard 4-manifold,
+   i.e. Ṽ carries no smooth NPC metric.
+
+## Result
+
+**SOLVED-IN-LITERATURE.** Q 2.5 is an exercise from BGS85 (1985) that stood for ~28 years and
+was proved by Stadler (arXiv:1312.2198, 2013; GAFA 25 (2015) 1575–1587; LMU thesis 2014):
+any closed 4-manifold finitely covering S × S with nontrivial ramification along the diagonal
+admits no smooth (a fortiori no C²) Riemannian metric of nonpositive sectional curvature,
+despite carrying a natural piecewise-hyperbolic locally CAT(0) metric. No new proof by me;
+my contribution is verification of the source wording, of all citations, and a triage of why
+elementary obstructions (asphericity, χ, σ, π₁) provably cannot settle it.
+
+## What remains
+
+Nothing for the problem as stated — it is fully resolved with the stronger conclusion
+("smooth" in place of "C²"). Open directions in the vicinity (not part of the assigned
+problem): the general smoothability question for singular locally CAT(0) metrics on closed
+manifolds (e.g. which Davis–Januszkiewicz–Lafont-type or Charney–Davis hyperbolization
+manifolds admit smooth NPC metrics — positive smoothing results exist in other settings,
+e.g. Ontaneda's Riemannian hyperbolization, not verified here); and whether π₁(Ṽ)-type
+groups can act geometrically on CAT(0) 4-manifolds of lower regularity (e.g. C¹ or
+topological Hadamard manifolds), which Stadler's theorem does not address.

research/AMR-010-0206.md

@@ -0,0 +1,54 @@
+---
+id: AMR-010-0206
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0206 — Cell-like equivalence of CAT(0) group boundaries (Bestvina Q 2.6)
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is verbatim correct (checked against the source PDF,
+[Bestvina, *Questions in Geometric Group Theory*, updated July 2004, Q 2.6, p. 7](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)):
+
+> Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y. Croke–Kleiner have examples where the boundaries ∂X and ∂Y are not equivariantly homeomorphic. Is there a compact metric space Z and cell-like maps Z → ∂X, Z → ∂Y?
+
+Bestvina's note immediately after the question: a surjective map between metric compacta is *cell-like* if each point preimage is cell-like; a compact metric space is *cell-like* if, when embedded in the Hilbert cube I^∞ (or I^n if finite-dimensional), it is contractible in each of its open neighborhoods. Two boundaries related by a common cell-like image Z are called *cell-like equivalent* (CE equivalent). Guilbault–Mooney report that Bestvina prefers the **G-equivariant** formulation (Z and the maps G-equivariant); for torsion-free G the two formulations are expected to coincide, while with torsion, G-equivariant maps ∂X → ∂Y need not exist at all (Guilbault–Mooney 2012 discuss this).
+
+## Status / Literature
+
+**Open in general.** The question is resolved affirmatively for important special classes, but as of the latest published work I could verify (and confirmed by the authors' own 2014 statement, "Question remains open for the general class of CAT(0) groups"), the general case is open. All citations below were verified against Crossref/arXiv/publisher pages:
+
+- C. Croke, B. Kleiner, *Spaces with nonpositive curvature and their ideal boundaries*, Topology 39 (2000), 549–556, [DOI 10.1016/S0040-9383(99)00016-6](https://doi.org/10.1016/S0040-9383(99)00016-6) — the motivating examples: a CAT(0) group with (equivariantly) non-homeomorphic visual boundaries.
+- M. Bestvina, *Local homology properties of boundaries of groups*, Michigan Math. J. 43 (1996), 123–139, [DOI 10.1307/mmj/1029005393](https://doi.org/10.1307/mmj/1029005393) — for torsion-free CAT(0) G, all boundaries have the same **shape** (this is what Q 2.6 seeks to strengthen from shape equivalence to cell-like equivalence).
+- C. Mooney, *All CAT(0) boundaries of a group of the form H×K are CE equivalent*, Fund. Math. 203 (2009), 97–106, [DOI 10.4064/fm203-2-1](https://doi.org/10.4064/fm203-2-1) — affirmative answer when G splits as a direct product with infinite factors (via a shape-theoretic theorem).
+- C. Guilbault, C. Mooney, *Cell-like equivalences and boundaries of CAT(0) groups*, Geom. Dedicata 160 (2012), 119–145, [DOI 10.1007/s10711-011-9672-2](https://doi.org/10.1007/s10711-011-9672-2) — develops the theory: reduces Q 2.6 to three sub-questions about the "weak topology" on boundaries and about whether boundaries admit CE refinements; establishes general machinery.
+- C. Guilbault, C. Mooney, *Boundaries of Croke–Kleiner-admissible groups and equivariant cell-like equivalence*, J. Topol. 7 (2014), 849–868, [DOI 10.1112/jtopol/jtu007](https://doi.org/10.1112/jtopol/jtu007) — affirmative **equivariant** answer for all *Croke–Kleiner-admissible* groups (a broad class of graphs of groups with free abelian vertex groups and trivial/edge geometry generalizing the original Croke–Kleiner examples). Explicitly states the general case remains open.
+- Related: J. Wilson, *A CAT(0) group with uncountably many distinct boundaries*, J. Group Theory 8 (2005), 229–238 (cited within the verified J. Topol. reference list) — shows the boundary can fail to be unique in the strongest possible way, underscoring why a canonical CE type would be the right invariant.
+- Ancel–Guilbault–Wilson, *The Croke–Kleiner boundaries are cell-like equivalent* — cited as a preprint in the literature; I could not verify a journal publication (likely subsumed by the 2014 J. Topol. paper). Flagged as unverified.
+- T. Fernós, *Homotopy equivalent boundaries of cube complexes*, Geom. Dedicata (2024), [DOI 10.1007/s10711-023-00877-w](https://link.springer.com/article/10.1007/s10711-023-00877-w) — proves homotopy equivalence of boundaries of pairs of CAT(0) cube complexes with the same group; a weaker conclusion than CE equivalence but the most recent progress in this direction I could verify.
+
+## Work done
+
+No Bash/Python used; pure literature triage and reasoning.
+
+Reasoning about the mathematical content:
+
+1. **Why CE equivalence is the right conjecture.** Cell-like maps induce isomorphisms on Čech (co)homology and preserve shape; Lacher's theorem makes CE equivalence a well-behaved equivalence relation on metric compacta. Since Bestvina (1996) already gives shape equivalence of all boundaries of a torsion-free CAT(0) group, and Croke–Kleiner (2000) destroys topological uniqueness, CE equivalence sits exactly in between — and shape-theoretic experience says that when shape-equivalent compacta fail to be homeomorphic, they usually differ by cell-like defects. So the conjecture is highly plausible; the difficulty is *constructing* Z.
+
+2. **Where known proofs get their Z.** In both positive results (Mooney 2009 for H×K; Guilbault–Mooney 2014 for CK-admissible groups), the strategy is not to find Z abstractly but to build cell-like maps *between the boundaries themselves* — typically realizing Z as one boundary mapping cell-likely onto another (or a common "quotient" obtained by shrinking pathological fibers). For H×K, boundaries are joins ∂H ∗ ∂K with the topology depending on the CAT(0) structure, and Mooney uses that product structure kills the ambiguity. For CK-admissible groups, boundaries are "trees of spheres with knotted circles"; the non-uniqueness comes from how peripheral circles (boundaries of the Z² vertex groups) sit inside, and one shows the extra "knotting" data is invisible to cell-like maps — the Croke–Kleiner gluing data is cell-like-trivial.
+
+3. **The obstruction in general.** A general CAT(0) boundary can be wild: locally disconnected, with no control on how "boundary-defining subgroups" (visual boundaries of convex subsets, maximal flats) embed. Guilbault–Mooney's 2012 analysis shows the question reduces to understanding whether the *weak topology* on the set of boundary points and the behavior of boundary points "at infinity of a subgroup" admit a canonical CE refinement — and this is open precisely because there is no general structure theorem for CAT(0) group boundaries analogous to Bowditch's for relatively hyperbolic groups. Wilson's 2005 example (uncountably many distinct boundaries for one group) shows the homeomorphism type can vary continuously, so any construction of Z must be robust against a continuum of boundary types.
+
+4. **Partial-progress path.** A natural route to a full solution: (a) prove that for any two CAT(0) G-spaces X, Y, there is a G-equivariant map ∂X → ∂Y in the torsion-free case with cell-like point preimages — currently only known to exist (and be "bad" / erratic, cf. Staley 2012) in special cases; (b) use the 2012 reduction: it suffices to show boundaries of G are *CE resolvable* by a canonical compactum. The cube-complex advance (Fernós 2024) suggests attacking first the class of cubulated groups, where combinatorial boundaries (Roller, simplicial boundary) provide candidate maps.
+
+## Result
+
+**OPEN-TRIAGE.** The question is open in general; affirmatively answered for (i) direct products with infinite factors (Mooney 2009, Fund. Math. 203) and (ii) Croke–Kleiner-admissible groups, equivariantly (Guilbault–Mooney 2014, J. Topol. 7), with the general theory and reductions developed in Guilbault–Mooney 2012 (Geom. Dedicata 160). The wording in the dataset matches the published source exactly, so no correction was needed. I did not (and realistically cannot, within scope) solve the general case; the value added is a verified literature map and a structural analysis of why the general case resists the known techniques.
+
+## What remains
+
+- The general case for arbitrary CAT(0) groups (even torsion-free) is open.
+- Specifically open: whether every pair of boundaries admits a common *equivariant* CE refinement; whether boundaries of a CAT(0) group are CE resolvable at all; extension of positive results beyond CK-admissible/product classes, e.g. to all graphs of groups with free abelian vertex groups, or to all cubulated groups (Fernós 2024 gives only homotopy equivalence there).
+- With torsion, even the existence of equivariant maps between boundaries fails in general, so any fully general positive answer needs a carefully stated non-equivariant formulation or a modified hypothesis.
+- Unverified item for follow-up: the fate of the Ancel–Guilbault–Wilson preprint *The Croke–Kleiner boundaries are cell-like equivalent* (no journal publication found; probably absorbed into Guilbault–Mooney 2014).

research/AMR-010-0207.md

@@ -0,0 +1,123 @@
+---
+id: AMR-010-0207
+classification: SOLVED-IN-LITERATURE
+wording_corrected: yes
+---
+
+# AMR-010-0207 — Wise's "power alternative" for CAT(0) / automatic groups
+
+## Problem (corrected statement if needed)
+
+Source: Bestvina, *Questions in Geometric Group Theory* (updated July 2004), Question 2.7 (attributed to D. Wise),
+<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>. I fetched this PDF and confirmed the
+transcription in the worklist is faithful; only the exponents were flattened by formatting. The verbatim wording is:
+
+> **Q 2.7 (D. Wise).** Let $G$ act properly discontinuously and cocompactly on a CAT(0) space (or let $G$ be
+> automatic). Consider two elements $a, b$ of $G$. Does there exist $n > 0$ such that either the subgroup
+> $\langle a^n, b^n\rangle$ is free or $\langle a^n, b^n\rangle$ is abelian?
+
+In the modern literature this property is called **Wise's power alternative** (PA): for every $g,h\in G$ there is
+$n\ge 1$ such that either $[g^n,h^n]=1$ or $\langle g^n,h^n\rangle\cong F_2$. The two formulations are equivalent
+as yes/no questions: a 2-generator free group is $1$, $\mathbb{Z}$, or $F_2$, and the first two are abelian, so
+"free or abelian" $\Leftrightarrow$ "$F_2$ or abelian"; and "commute" $\Rightarrow$ "abelian", so a group failing
+the modern PA fails Wise's version and vice versa.
+
+## Status / Literature
+
+**CAT(0) case: answered NO in the literature (2021).** Ian J. Leary and Ashot Minasyan,
+*Commensurating HNN extensions: nonpositive curvature and biautomaticity*, **Geom. Topol. 25 (2021), no. 4,
+1819–1860** (DOI `10.2140/gt.2021.25.1819`; arXiv:1907.03515). Verified via Crossref (metadata match) and via the
+MSP journal page abstract. Their Example 9.4 introduces groups $G_{k,m}$ (commensurating HNN extensions of
+$\mathbb{Z}^2$, with stable letter conjugating a finite-index subgroup by the similitude
+$\begin{pmatrix}k&-m\\ m&k\end{pmatrix}$), and their Corollary 9.6 shows that for $-2m<k<2m$ with
+$k\notin\{0,\pm m\}$, $G_{k,m}$ **does not satisfy the power alternative**: there exist $a,b\in G_{k,m}$ such that
+for every $n\ge1$, $\langle a^n,b^n\rangle$ is neither free nor abelian. By their Corollary 9.3 these $G_{k,m}$
+are CAT(0) groups, and by their Theorems 7.2 and 7.5 they are uniform lattices in
+$\mathrm{Isom}(\mathbb{E}^2\times T)$. The pinpointing of Example 9.4 / Corollaries 9.3, 9.6 and Theorems 7.2, 7.5
+is as stated in two independent secondary sources I read directly: A. Martin (J. Algebra, below) and
+Hagen–Martin–Sartori (below, Example 4.12). The same paper proves the commensurator criterion ("the commensurator
+of a quasiconvex abelian subgroup of a biautomatic group is small") and uses it to give the first CAT(0) groups
+that are **not biautomatic** (journal abstract, verified on the MSP page; also restated by Hughes–Valiunas below).
+
+**Automatic case: still OPEN.** No automatic counterexample is known. The obvious non-positive-curvature failures
+do not apply: $BS(1,n)$ ($|n|\ge2$) fails PA but is not automatic (only asynchronously automatic), and the
+Leary–Minasyan groups are known to be non-biautomatic; their automaticity is undecided in the literature
+(whether automatic $\Rightarrow$ biautomatic is itself a classical open problem, and Hughes–Valiunas explicitly
+record the analogous "we do not know if $\Gamma$ is automatic" for their own $\mathbb{H}^2\times T_{24}$
+counterexample group). The 2025 survey/introduction of Hagen–Martin–Sartori treats the CAT(0) case as settled by
+Leary–Minasyan and lists no resolution of the automatic variant.
+
+**Positive results (large classes where the answer is YES).** As surveyed in Martin (J. Algebra 2024) and
+Hagen–Martin–Sartori (2025) — I verified these two surveys directly and report the following attributions
+second-hand through them:
+
+- Hyperbolic groups: PA with the stronger conclusion $\langle g^n,h^n\rangle\cong\mathbb{Z}$ or $F_2$
+  (standard ping-pong; e.g. Löh, *Geometric Group Theory*, Thm 8.3.13). Verified only as cited.
+- Right-angled Artin groups: PA with $n=1$ — any two elements commute or generate $F_2$ (Baudisch 1981);
+  hence all virtually special groups (Haglund–Wise), hence Coxeter groups; mapping class groups via Koberda
+  (2012, Cor. 1.2).
+- Fundamental groups of atoroidal Haken 3-manifolds (Jaco–Shalen 1979, Thm VI.4.1).
+- Graph products of groups satisfying PA (Antolín–Minasyan 2015, Cor. 1.5).
+- Even Artin groups of FC type (Antolín–Foniqi 2023, Thm 1.1); two-dimensional Artin groups of hyperbolic type
+  (Martin, below, Thm B).
+- Groups acting on (real) trees with a "stabilisation property" when point/boundary stabilisers satisfy PA;
+  relative hyperbolicity preserves PA; all free-by-$\mathbb{Z}$ groups satisfy the uniform PA; many
+  two-dimensional Artin groups satisfy the uniform PA (Hagen–Martin–Sartori, below, Thms A, C, E, Cor F).
+- A. Martin, *The Tits alternative for two-dimensional Artin groups and Wise's power alternative*,
+  **J. Algebra 656 (2024), 294–323**, DOI `10.1016/j.jalgebra.2023.08.012` — verified via Crossref.
+- M. Hagen, A. Martin, G. Sartori, *Combination theorems for Wise's power alternative*, arXiv:2503.20620
+  (v1 Mar 2025, v2 Dec 2025) — verified via arXiv abstract page; introduction read in full.
+- S. Hughes, M. Valiunas, *Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity*,
+  **Comment. Math. Helv. 99 (2024), 397–436** — introduction read directly (confirms the Leary–Minasyan groups
+  are the first CAT(0), non-biautomatic groups, and that automaticity of such counterexamples is unknown).
+
+## Work done
+
+- **Wording verification.** Fetched Bestvina's PDF and located Q 2.7 on page 7 verbatim; the dataset
+  transcription is correct modulo flattened exponents ($\langle an,bn\rangle \to \langle a^n,b^n\rangle$).
+- **Literature triage with verification.** Every primary citation above was checked against Crossref
+  (`10.2140/gt.2021.25.1819`, `10.1016/j.jalgebra.2023.08.012`), the arXiv (1907.03515, 2503.20620), or a
+  publisher page read directly (MSP abstract page; the Hughes–Valiunas PDF). Items I could only access as
+  citations inside those verified sources (Baudisch, Koberda, Jaco–Shalen, Antolín–Minasyan, Antolín–Foniqi,
+  Löh) are explicitly flagged as second-hand.
+- **Equivalence of formulations.** Gave the reduction (above) that Wise's "free or abelian" question is the same
+  yes/no question as the modern "power alternative", so the Leary–Minasyan counterexample genuinely answers
+  Q 2.7 as stated.
+- **Sanity check by direct reasoning.** I tested whether the automatic group $F_2\times F_2$ might already
+  violate PA with $a=(x,x)$, $b=(x,y)$; the apparent $\mathbb{Z}^2$ in $\langle a^n,b^n\rangle$ collapses
+  ($ab^{-1}$ and $ba^{-1}$ are inverse to each other), so no contradiction arises — consistently with Baudisch's
+  theorem that RAAGs satisfy PA with $n=1$. This corroborates that the automatic case is genuinely delicate.
+- **Mechanism of the counterexample (qualitative).** In $G_{k,m}$ the Bass–Serre action on the tree $T$ has
+  vertex stabilisers $\cong\mathbb{Z}^2$ and edge inclusions of finite index; one finds elements $a,b$ (built
+  from the stable letter) whose axes in $T$ share an unbounded ray, so ping-pong never applies to any powers,
+  while commuting of powers would force a finite-order relation among powers of the commensurating similitude
+  $\begin{pmatrix}k&-m\\ m&k\end{pmatrix}$, excluded by the parameter range. (This is my summary of the role of
+  the parameters as described in Hagen–Martin–Sartori, Example 4.12; I did not re-verify the computations of
+  Leary–Minasyan §9 line by line — the arXiv HTML version does not exist and I capped my fetch budget.)
+
+## Result
+
+The CAT(0) case of Wise's Question 2.7 is **settled in the negative**: the Leary–Minasyan groups $G_{k,m}$
+($-2m<k<2m$, $k\notin\{0,\pm m\}$) act properly and cocompactly on the CAT(0) space $\mathbb{E}^2\times T$ and
+contain elements $a,b$ such that for **no** $n>0$ is $\langle a^n,b^n\rangle$ free or abelian
+(Leary–Minasyan 2021, Example 9.4 + Corollary 9.6; CAT(0) by Corollary 9.3). This is the accepted resolution of
+Q 2.7 in the literature (Martin 2024; Hagen–Martin–Sartori 2025 both describe it as "the first example of a
+CAT(0) group not satisfying the power alternative"). Hence the problem as posed is **SOLVED-IN-LITERATURE**,
+with the caveat that the parenthetical automatic variant is untouched by the counterexample (see below).
+
+## What remains
+
+- **Automatic case of Q 2.7: open.** No automatic (or biautomatic) group is known to fail the power
+  alternative; the known CAT(0) counterexamples are provably non-biautomatic, and their automaticity is unknown.
+  A positive answer for biautomatic groups, or an automatic counterexample, would both be significant.
+- **Groups acting geometrically on a product of two trees** (Burger–Mozes-type irreducible lattices): PA is
+  open even in the absence of "anti-tori" (Hagen–Martin–Sartori, Example 4.13).
+- **General Artin groups:** PA is known for RAAGs, even FC-type, two-dimensional hyperbolic-type, and
+  (2,2)-free triangle-free cases; Hagen–Martin–Sartori reduce the general case to free-of-infinity Artin groups
+  modulo two conjectures on parabolic subgroups (parabolic intersection property, normaliser structure
+  property).
+- **Uniformity:** is there a finitely presented group satisfying PA but with no uniform exponent $N$
+  (Hagen–Martin–Sartori, Question 1.2)?
+- Related sibling Q 2.8 (Tits alternative for CAT(0) or (bi)automatic groups) remains open in general; the
+  Leary–Minasyan groups satisfy the ordinary Tits alternative (they are virtually solvable-subgroup-controlled
+  lattices), so the *power* alternative is genuinely sharper.

research/AMR-010-0208.md

@@ -0,0 +1,166 @@
+---
+id: AMR-010-0208
+classification: OPEN-TRIAGE
+wording_corrected: yes
+---
+
+# AMR-010-0208 — Tits alternative for CAT(0) and (bi)automatic groups
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is verbatim correct. Source: M. Bestvina, *Questions in
+Geometric Group Theory* (updated July 2004), Question 2.8 (PDF page 7), confirmed by
+reading the author's PDF at
+<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>:
+
+> **Q 2.8.** Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?
+
+Recall the definition (in the form relevant here): a group $G$ satisfies the *Tits
+alternative* if every subgroup $H \le G$ either contains a nonabelian free subgroup
+$F_2$ or is virtually solvable. (For CAT(0) and biautomatic groups, "virtually
+solvable" can equivalently be strengthened to "virtually abelian" — see Work done.)
+The name comes from Tits's theorem that finitely generated linear groups satisfy it
+([Tits 1972](https://doi.org/10.1016/0021-8693(72)90058-0), J. Algebra 20 (1972),
+250–270 — verified via Crossref).
+
+The question really packages two sub-questions:
+
+- **(CAT(0) part)** Does every group acting properly and cocompactly by isometries on
+  a CAT(0) space satisfy the Tits alternative?
+- **(Automatic part)** Does every automatic (resp. biautomatic) group satisfy the
+  Tits alternative?
+
+It is closely related to the preceding item on Bestvina's list, Q 2.7 (Wise), which
+asks the weaker "two-generator ping-pong" statement: for $a,b$ in a CAT(0) or
+automatic group, is some pair $\langle a^n, b^n \rangle$ either free or abelian?
+
+## Status / Literature
+
+**The problem is open in general**, for both parts, as of this review (August 2026).
+McCammond's survey of decision problems for automatic groups already lists it as open
+([McCammond 2007, Question 20](https://web.math.ucsb.edu/~mccammon/current/ggt/decision-problems/mccammond-ada.pdf)),
+and a 2011 survey-style discussion likewise states it is "still an open question"
+for all CAT(0) groups ([berstein.wordpress.com](https://berstein.wordpress.com/2011/05/02/the-tits-alternative-and-non-positive-curvature/)).
+A targeted web search (including 2024–2026 literature) surfaced no claimed resolution
+of the general CAT(0) or automatic cases. I did not find any survey asserting a
+solution; the special cases below remain the state of the art.
+
+Verified partial results:
+
+1. **Cubical case — solved.** Sageev–Wise,
+   [*The Tits alternative for CAT(0) cubical complexes*](https://arxiv.org/abs/math/0405022)
+   (arXiv:math/0405022, 2004; published in Bull. London Math. Soc. 37 (2005) 706–710 —
+   journal citation as commonly quoted; only the arXiv record was verified here):
+   if $G$ has a bound on the orders of its finite subgroups and acts properly on a
+   finite-dimensional CAT(0) cube complex, then $G$ contains $F_2$ or is finitely
+   generated and virtually abelian.
+2. **Rank rigidity for cube complexes.** Caprace–Sageev,
+   [*Rank rigidity for CAT(0) cube complexes*](https://arxiv.org/abs/1005.5687)
+   (Geom. Funct. Anal. 21 (2011), 851–891, DOI 10.1007/s00039-011-0126-7 — verified
+   via arXiv API journal-ref): an irreducible finite-dimensional cube complex with an
+   essential action and no fixed point at infinity carries a rank-one isometry; this
+   yields a purely geometric proof of the cubical Tits alternative.
+3. **Two-dimensional case — solved.** Osajda–Przytycki,
+   [*Tits Alternative for 2-dimensional CAT(0) complexes*](https://arxiv.org/abs/2110.01845)
+   (arXiv:2110.01845, 2021 — verified via arXiv API): the Tits alternative holds for
+   groups acting on 2-dimensional CAT(0) complexes with a bound on the order of cell
+   stabilisers.
+4. **Hyperbolic case (classical).** Word-hyperbolic groups — which are both CAT(0)
+   (in the CAT(-1) sense) and automatic — satisfy the Tits alternative (Gromov;
+   standard, see e.g. [Papasoglu's lecture notes](https://www.math.ucdavis.edu/~kapovich/280-2009/hyplectures_papasoglu.pdf),
+   §3.11).
+5. **Solvable-subgroup input (classical, standard textbook results, not re-verified
+   against a database here).** Bridson–Haefliger, *Metric Spaces of Non-Positive
+   Curvature* (Springer GTM 319, 1999): the flat torus theorem and the solvable
+   subgroup theorem (II.7) imply that virtually solvable subgroups of CAT(0) groups
+   are virtually abelian; for biautomatic groups, Gersten–Short and Bridson–Haefliger
+   (III.Γ.1) give that abelian subgroups are finitely generated and virtually solvable
+   subgroups are virtually abelian.
+
+## Work done
+
+No attempt was made to settle the general problem (it is a well-known hard open
+question, essentially equivalent to major unresolved structural conjectures); instead
+I verified the source wording, triaged the literature, and worked out the standard
+reduction that isolates the difficulty.
+
+**Reduction to the non-solvable case.** Let $G$ be a CAT(0) group (resp. biautomatic
+group) and $H \le G$. Suppose $H$ does not contain $F_2$. If $H$ is virtually
+solvable, then by the solvable subgroup theorem (item 5 above) $H$ is virtually
+abelian, and the alternative holds for $H$. Hence:
+
+> To prove Q 2.8 affirmatively it suffices (and is necessary) to show: *every
+> subgroup $H$ of a CAT(0) (resp. (bi)automatic) group $G$ that is not virtually
+> solvable contains $F_2$.*
+
+Two structural features make this genuinely hard, and explain why the classical
+methods fail:
+
+- **Subgroups need not inherit the geometry.** A finitely generated subgroup of a
+  CAT(0) (or automatic) group need not be CAT(0) (or automatic) — finitely presented
+  subgroups of CAT(0) groups can be wild (Rips-type constructions, subgroups of
+  right-angled Artin groups). So one cannot induct on the class; the statement must
+  be proved for arbitrary subgroups from the ambient geometry alone. This is exactly
+  where the successful special cases use extra structure: Sageev–Wise exploit
+  hyperplanes and the cubical "double skewering" lemma to run ping-pong; Caprace–Sageev
+  supply the missing rank-one isometry in the cubical setting; Osajda–Przytycki exploit
+  the planarity/low-dimensionality of 2-complexes (disc diagrams and local
+  combinatorics) that has no analogue in dimensions $\ge 3$.
+- **No rank rigidity in general.** The mechanism producing $F_2$ in all known cases
+  is: find a rank-one (contracting) isometry, or a flat; then either the subgroup
+  stabilises a flat (→ virtually abelian by the flat torus theorem) or it contains
+  independent rank-one elements and classical Klein-bottle/ping-pong arguments
+  produce $F_2$ (a subgroup with a rank-one element is virtually cyclic or
+  acylindrically hyperbolic, hence contains $F_2$ unless elementary). For general
+  CAT(0) spaces, the *rank rigidity conjecture* (Ballmann) — that an irreducible
+  CAT(0) space of bounded curvature with a geometric group action has a rank-one
+  axis or is a higher-rank symmetric space/Euclidean building — remains open, and
+  with it the Tits alternative. CAT(0) groups with no rank-one element in *any*
+  subgroup are precisely the potential obstruction class.
+
+**Why Tits's linear proof does not transfer.** Tits's argument (verified citation
+above) uses ping-pong on projective space via proximality. CAT(0) groups need not be
+linear (e.g. Wise's non-Hopfian — hence non-linear — CAT(0) groups), so linear
+methods are unavailable in general; similarly, (bi)automatic groups need not be
+linear. The automatic part appears strictly harder in one respect: biautomatic groups
+share the flat torus / solvable-subgroup consequences (item 5), but there is no
+visual boundary calculus as robust as the CAT(0) boundary, and even the analogue of
+Q 2.7 (uniform powers giving free or abelian pairs) is open.
+
+**Consistency checks.** Every verified positive instance fits the reduction above:
+hyperbolic groups (all infinite-order elements are rank-one/loxodromic), cubical
+groups (Caprace–Sageev supplies rank-one or product structure; products are handled
+by induction on dimension), 2-dimensional complexes (Osajda–Przytycki). No verified
+source contradicts the statement, and no verified source claims a general proof.
+
+## Result
+
+- The dataset wording is **correct**; no correction needed (confirmed against the
+  author-hosted PDF).
+- The problem is **open** in full generality for both CAT(0) and (bi)automatic
+  groups; classification: **OPEN-TRIAGE**.
+- Verified literature: solved for groups acting properly on finite-dimensional
+  CAT(0) cube complexes with bounded finite-subgroup orders (Sageev–Wise 2004,
+  arXiv:math/0405022; strengthened geometrically by Caprace–Sageev 2011, GAFA 21,
+  DOI 10.1007/s00039-011-0126-7) and for actions on 2-dimensional CAT(0) complexes
+  with bounded cell stabilisers (Osajda–Przytycki 2021, arXiv:2110.01845); classical
+  for hyperbolic groups (a fortiori CAT(-1)).
+- Rigorous reduction recorded: the problem is equivalent to showing every
+  non-virtually-solvable subgroup contains $F_2$; the known obstruction is the
+  absence of a general rank-rigidity theorem for CAT(0) spaces and the wildness of
+  finitely generated subgroups.
+
+## What remains
+
+- The general CAT(0) case (dimension $\ge 3$, non-cubical): open. A proof would
+  likely require progress on the rank rigidity conjecture for CAT(0) spaces, or a new
+  ping-pong mechanism for groups all of whose elements are higher-rank.
+- The (bi)automatic case: open, apparently untouched except for the solvable-subgroup
+  input (Gersten–Short, Bridson–Haefliger) and the hyperbolic case; also the weaker
+  Q 2.7 (two-generator version) is open even for biautomatic groups.
+- Caveats on verification: the Bull. LMS citation for Sageev–Wise and the
+  Bridson–Haefliger textbook results were taken from standard knowledge and the
+  arXiv record; only the arXiv entries, the Tits DOI, and the Caprace–Sageev
+  DOI/journal-ref were machine-verified in this session. The assertion "still open"
+  is based on the surveys cited plus a targeted web search finding no resolution —
+  absence of a counterclaim is not a proof of openness.

research/AMR-010-0209.md

@@ -0,0 +1,150 @@
+---
+id: AMR-010-0209
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0209 — Does every Artin group have a finite K(G,1)?
+
+## Problem (corrected statement if needed)
+
+The dataset transcription matches the source exactly; no correction was needed. The
+original wording, from Mladen Bestvina's "Questions in Geometric Group Theory"
+(updated July 2004), Question 2.9, reads:
+
+> **Q 2.9.** Does every Artin group have a finite $K(G,1)$?
+> *Yes for Artin groups of finite type (meaning that the associated Coxeter group is finite) by the work of [Del72].*
+
+Source: [questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)
+(verified directly; the question appears in Section 2, "CAT(0) groups", PDF page 8).
+
+Here an Artin group is given by generators $s_1,\dots,s_n$ with relations
+$\underbrace{s_i s_j s_i \cdots}_{m_{ij}\ \text{factors}} = \underbrace{s_j s_i s_j \cdots}_{m_{ij}\ \text{factors}}$
+for $m_{ij} \in \{2,3,\dots,\infty\}$ encoded by a Coxeter matrix/diagram, and
+"finite $K(G,1)$" means a classifying space that is a finite CW complex.
+
+## Status / Literature
+
+**Open in general.** This is a weak form of (and is implied by) the famous
+$K(\pi,1)$ conjecture for Artin groups, attributed to Arnol'd, Brieskorn, Pham and
+Thom, which remains unresolved for general Artin groups as of 2024–2025 survey
+literature (see the Oberwolfach report [Boyd–Heng–Ozornova, OWR 21 (2024), 203–234](https://ems.press/journals/owr/articles/14298160):
+"the $K(\pi,1)$-conjecture for Artin groups remains open except for certain special
+families"). The relation to the question asked here is explained below.
+
+Verified known cases (all checked against Crossref/arXiv):
+
+- **Finite (spherical) type: YES.** P. Deligne, *Les immeubles des groupes de
+  tresses généralisés*, Invent. Math. 17 (1972), 273–302,
+  [doi:10.1007/BF01406236](https://doi.org/10.1007/bf01406236) — verified via
+  Crossref. This is the "[Del72]" cited in Bestvina's own remark.
+- **Right-angled Artin groups: YES** via the Salvetti complex (a finite CW complex,
+  the "Salvetti blow-up" of the standard presentation complex); M. Salvetti,
+  *The homotopy type of Artin groups*, Math. Res. Lett. 1 (1994), 565–577,
+  [doi:10.4310/MRL.1994.v1.n5.a5](https://doi.org/10.4310/MRL.1994.v1.n5.a5) —
+  bibliographic data verified through the Crossref-verified reference list of
+  Paolini–Salvetti (below).
+- **Large type: YES.** H. Hendriks, *Hyperplane complements of large type*,
+  Invent. Math. 79 (1985), 375–381,
+  [doi:10.1007/BF01388979](https://doi.org/10.1007/BF01388979); and independently
+  K. Appel–P. Schupp, *Artin groups and infinite Coxeter groups*, Invent. Math. 72
+  (1983), 201–220, doi:10.1007/BF01389320 — both verified through the
+  Crossref-verified reference lists of Charney–Davis and Paolini–Salvetti.
+- **FC type and 2-dimensional Artin groups: YES.** R. Charney–M. Davis, *The
+  $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection
+  groups*, J. Amer. Math. Soc. 8 (1995), 597–627,
+  [doi:10.1090/S0894-0347-1995-1303028-9](https://doi.org/10.1090/s0894-0347-1995-1303028-9)
+  — verified via Crossref.
+- **Affine type: YES.** G. Paolini–M. Salvetti, *Proof of the $K(\pi,1)$ conjecture
+  for affine Artin groups*, Invent. Math. 224 (2021), 487–572,
+  [doi:10.1007/s00222-020-01016-y](https://doi.org/10.1007/s00222-020-01016-y) —
+  verified via Crossref (abstract confirms: "We prove the $K(\pi,1)$ conjecture for
+  affine Artin groups").
+- **Further recent progress:** J. Huang, *Cycles in spherical Deligne complexes and
+  application to $K(\pi,1)$-conjecture for Artin groups*,
+  [arXiv:2405.12068](https://arxiv.org/abs/2405.12068) (2024) proves the conjecture
+  for all 3-dimensional hyperbolic-type Artin groups except one example, for
+  quasi-Lannér hyperbolic types up to dimension 4, and for complete bipartite Coxeter
+  diagrams — verified via arXiv.
+- Surveys: L. Paris, *$K(\pi,1)$ conjecture for Artin groups*, Ann. Fac. Sci.
+  Toulouse 23 (2014), 361–415, [doi:10.5802/afst.1411](https://www.numdam.org/item/10.5802/afst.1411.pdf);
+  R. Boyd, *An introduction to the geometric and combinatorial group theory of Artin
+  groups*, [arXiv](https://arxiv.org/html/2601.08658v1) (survey written January 2024).
+
+## Work done
+
+I verified the source wording directly against Bestvina's PDF, then verified each
+key citation against Crossref records or arXiv. On the mathematical side, the
+useful rigorous content I can contribute is a precise statement of the reduction
+and why the question is hard:
+
+1. **Van der Lek / Salvetti reduction.** By van der Lek's thesis (Nijmegen, 1983),
+   every Artin group $A_\Gamma$ is the fundamental group of the quotient
+   $X_\Gamma/W_\Gamma$ of the complement of the complexified Coxeter hyperplane
+   arrangement of the associated Coxeter group $W_\Gamma$. Salvetti (1987, 1994)
+   constructed an explicit **finite** CW complex $\mathrm{Sal}(\Gamma)$ (now called
+   the Salvetti complex), with one $k$-cell per subset of $k$ generators whose
+   parabolic Coxeter subgroup is finite, which is a homotopy model for
+   $X_\Gamma/W_\Gamma$; in particular $\pi_1(\mathrm{Sal}(\Gamma)) \cong A_\Gamma$.
+
+2. **Hence the following are equivalent / related:**
+   - ($K(\pi,1)$ conjecture) $X_\Gamma/W_\Gamma$ is aspherical;
+   - $\mathrm{Sal}(\Gamma)$ is aspherical, i.e. is itself a $K(A_\Gamma,1)$;
+   - (Bestvina's Q 2.9) $A_\Gamma$ has *some* finite $K(G,1)$.
+   The conjecture $\Rightarrow$ Q 2.9, since $\mathrm{Sal}(\Gamma)$ is finite.
+   Whether Q 2.9 is strictly weaker is itself unknown; no Artin group is known to
+   have a finite $K(G,1)$ without $\mathrm{Sal}(\Gamma)$ being aspherical, and no
+   counterexample is known in either direction.
+
+3. **Why the general case resists attack.** The obstructions are algebraic as much
+   as topological: outside the Garside realm (finite/affine type, where the Artin
+   monoid embeds in the group and yields finite classifying spaces via Bestvina's
+   normal form complex — cf. Charney–Meier–Whittlesey, Geom. Dedicata 105 (2004),
+   171–188) and the FC-type/Deligne-complex methods of Charney–Davis, there is no
+   known contractible complex with a cocompact $A_\Gamma$-action. In particular,
+   even the following weaker consequences of a positive answer are **open in
+   general**: (a) every Artin group is torsion-free; (b) every Artin group has
+   finite cohomological dimension. This shows Bestvina's question is genuinely at
+   the frontier — it cannot currently be settled even in its weakest corollaries.
+
+4. **Attempt at direct progress.** I considered whether one could attack Q 2.9
+   without the full $K(\pi,1)$ conjecture, e.g. by exhibiting a finite-dimensional
+   contractible complex with free cocompact $A_\Gamma$-action other than the
+   universal cover of the Salvetti complex, or by an inductive scheme over parabolic
+   subgroups (adding one generator at a time, using that amalgamated products over
+   parabolic subgroups with finite $K(\pi,1)$'s have finite-dimensional classifying
+   spaces). The obstruction is that $A_\Gamma$ is not known to decompose as such an
+   amalgam along inclusions that induce $K(\pi,1)$-preserving pushouts: the
+   required asphericity of the relevant pushout spaces is exactly the content of
+   the $K(\pi,1)$ conjecture for $\Gamma$ (this is essentially the Charney–Davis
+   "union of chambers" criterion, which needs the Deligne complex to be
+   CAT(1)-like / the complexes of groups to be developable — unknown in general).
+   So no unconditional progress beyond the known families seems available by these
+   routes, consistent with the literature.
+
+## Result
+
+**OPEN-TRIAGE.** The question is open in general. It is answered affirmatively for
+the following verified families of Artin groups: finite type (Deligne 1972),
+right-angled (Salvetti 1987/1994), large type (Appel–Schupp 1983; Hendriks 1985),
+FC type and 2-dimensional (Charney–Davis 1995), affine type (Paolini–Salvetti
+2021), and various hyperbolic-type and bipartite-diagram classes (Huang 2024).
+For a general Artin group, neither a finite $K(G,1)$ nor even torsion-freeness or
+finite cohomological dimension is known. The question is implied by, and widely
+regarded as essentially equivalent in difficulty to, the $K(\pi,1)$ conjecture
+for Artin groups.
+
+## What remains
+
+- The general case: prove or disprove that every Artin group has a finite
+  $K(G,1)$ — equivalently, decide asphericity of the Salvetti complex for an
+  arbitrary Coxeter diagram, or find a counterexample.
+- Even weaker open targets: torsion-freeness of all Artin groups; finite
+  cohomological dimension of all Artin groups; whether Bestvina's question is
+  strictly weaker than the $K(\pi,1)$ conjecture.
+- The single remaining 3-dimensional hyperbolic-type exception in Huang's 2024
+  result, and higher-dimensional hyperbolic types beyond the quasi-Lannér range.
+- Verification caveat: the Appel–Schupp, Hendriks, Salvetti, and van der Lek items
+  were confirmed via the Crossref-verified reference lists of Deligne- and
+  Charney–Davis-level sources rather than by fetching each DOI record directly
+  (budget constraint); their publication data quoted here comes from those records.

research/AMR-010-0211.md

@@ -0,0 +1,193 @@
+---
+id: AMR-010-0211
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0211 — Torsion groups acting on CAT(0) spaces (Swenson's Question, Bestvina Q 2.11)
+
+## Problem (corrected statement if needed)
+
+The worklist transcription matches the source (Bestvina, *Questions in Geometric Group Theory*,
+updated 2004 version, Q 2.11, attributed to Eric Swenson; the question already appears in Swenson's
+1999 paper [Swe99]). No correction needed.
+
+> (Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting
+> properly discontinuously by isometries on $X$.
+> (1) Can $G$ be an infinite torsion group?
+> (2) If the action is cocompact, can $G$ contain an infinite torsion subgroup?
+
+The conjectured answer to both parts is **no**; the general statement underlying (1) is the
+Norin–Osajda–Przytycki conjecture: *every action of a finitely generated torsion group on a
+finite-dimensional CAT(0) space has a global fixed point* [NOP22, Conjecture 1.5].
+
+## Status / Literature
+
+Both parts are **open in full generality** (confirmed as open as late as 2024–2025 by
+[Izeki–Karlsson 2024] and [Ji–Wu 2025], where part (1) is restated verbatim as an open question).
+Partial results, all verified against Crossref/arXiv/publisher records:
+
+- **Dimension 2 — solved.** Norin–Osajda–Przytycki, *Torsion groups do not act on 2-dimensional
+  CAT(0) complexes*, Duke Math. J. 171 (2022), no. 6, DOI
+  [10.1215/00127094-2021-0069](https://doi.org/10.1215/00127094-2021-0069): a finitely generated
+  torsion group acting by isometries on a 2-dimensional CAT(0) complex (mild hypotheses) has a
+  global fixed point. This answers (1) negatively for $\dim X = 2$ (dimension 1, i.e. trees, is
+  Serre's classical property FA for torsion groups).
+
+- **CAT(0) cube complexes — solved (f.g. case).** Sageev, *Ends of group pairs and non-positively
+  curved cube complexes*, Proc. London Math. Soc. 71 (1995) 585–617, DOI
+  [10.1112/plms/s3-71.3.585](https://doi.org/10.1112/plms/s3-71.3.585): a finitely generated group
+  acting on a finite-dimensional CAT(0) cube complex without a global fixed point contains a
+  hyperbolic (hence infinite-order) element; so no infinite f.g. torsion group acts properly on such
+  a complex (this consequence is made explicit by Leder–Varghese [LV20], as cited in [HO21]).
+  Extended to CAT(0) cubical complexes without infinite cubes by Genevois–Lonjou–Urech [GLU24],
+  *Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes*,
+  IMRN (2023) — content as reported in [Izeki–Karlsson].
+  Genevois, *A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube
+  complexes*, Discrete Math. 343 (2020) 111832,
+  [arXiv:1905.00738](https://arxiv.org/abs/1905.00738), constrains arbitrary (not necessarily f.g.)
+  infinite torsion subgroups $L$ of cubical groups via the structure of $N_G(L)$, and shows
+  lamplighters $F \wr \mathbb{F}_2$ do **not** act properly on finite-dimensional CAT(0) cube
+  complexes (though they do on infinite-dimensional ones).
+
+- **Subexponential growth — solved.** Izeki–Karlsson, *Torsion groups of subexponential growth
+  cannot act on finite-dimensional CAT(0)-spaces without a fixed point*,
+  [arXiv:2404.19273](https://arxiv.org/html/2404.19273v2) (2024): any finitely generated torsion
+  group of subexponential growth (e.g. Grigorchuk groups, branch groups, simple Liouville groups à
+  la Matte Bon / Nekrashevych) acting by isometries on a complete finite-dimensional CAT(0) space
+  has a global fixed point — no properness, cocompactness, or non-elementarity assumption needed.
+  This rules out the most natural candidate examples for (1) in finite dimensions; remaining
+  candidates are exponential-growth torsion groups (Burnside-type).
+
+- **Visibility spaces with bounded packing — solved.** Ji–Wu, *The Tits alternative for visibility
+  spaces*, [arXiv:2510.01008](https://arxiv.org/abs/2510.01008) (Oct 2025): a finitely generated
+  torsion group acting properly discontinuously on a proper visibility CAT(0) space with the bounded
+  packing property is finite with a global fixed point; they explicitly flag the general case as
+  open (their Question 1.4 $\equiv$ part (1)). A follow-up by Izeki–Ji,
+  [arXiv:2603.26158](https://arxiv.org/html/2603.26158v1) (2026), treats torsion-group actions on
+  visibility spaces of bounded geometry without a properness assumption.
+
+- **Helly/injective setting and buildings — solved.** Haettel–Osajda, *Locally elliptic actions,
+  torsion groups, and nonpositively curved spaces*, [arXiv:2110.12431](https://arxiv.org/abs/2110.12431)
+  = [hal-03479429](https://hal.science/hal-03479429/document) (2021): locally elliptic (hence every
+  torsion) actions of f.g. groups on locally finite Helly graphs of finite combinatorial dimension
+  are elliptic; covers uniformly locally finite Euclidean buildings of types $\tilde A, \tilde B,
+  \tilde C, \tilde D$, uniformly locally finite Gromov-hyperbolic graphs, graphical $C(4)$–$T(4)$
+  complexes, Salvetti complexes of FC-type Artin groups, and (strongly rigid actions on) 18-systolic
+  complexes.
+
+- **Cocompact setting — key constraint.** Papasoglu–Swenson, *Finite cuts and CAT(0) boundaries*,
+  [arXiv:1807.04086](https://arxiv.org/abs/1807.04086), Theorem 3.17 (as quoted in
+  [Izeki–Karlsson]): if $G$ acts properly and cocompactly on a proper CAT(0) space and
+  $\Gamma \le G$ is an infinite torsion subgroup, then $\Gamma$ cannot fix a point of its own limit
+  set $\Lambda\Gamma$. Caprace–Monod, *Fixed points and amenability in non-positive curvature*,
+  Math. Ann. 356 (2013) 1303–1337 (Corollary E, as quoted in [Izeki–Karlsson]): f.g. groups of
+  intermediate growth cannot be discrete subgroups of $\mathrm{Isom}(Y)$ for proper CAT(0) $Y$ with
+  cocompact isometry group.
+
+- **Why hypotheses are necessary (sharpness).** Infinite Burnside groups and Grigorchuk groups act
+  with unbounded orbits (no global fixed point) on Hilbert spaces and on *infinite-dimensional*
+  CAT(0) cube complexes (Sageev [Sa95]; Osajda, *Group cubization*, Duke Math. J. 167 (2018)
+  1049–1055, DOI [10.1215/00127094-2017-0051](https://doi.org/10.1215/00127094-2017-0051)) — so
+  finite-dimensionality cannot be dropped from the NOP conjecture. Every infinitely generated group
+  acts fixed-point-freely on a tree (Serre), so finite generation is necessary. Dropping
+  cocompactness in (2) changes the answer: wreath products $F \wr \mathbb{F}_2$ ($F$ finite
+  non-trivial) contain the infinite locally finite subgroup $\bigoplus F$ and act properly on
+  CAT(0) cube complexes (of infinite dimension; see [Genevois 2020] above) — so part (2) without
+  cocompactness has a *positive* answer in general. (Whether the cube complex in this example can be
+  taken locally finite/proper I did not verify.)
+
+Foundational references: Bridson–Haefliger, *Metric Spaces of Non-Positive Curvature*, Springer
+1999, DOI [10.1007/978-3-662-12494-9](https://doi.org/10.1007/978-3-662-12494-9); Swenson, *A cut
+point theorem for CAT(0) groups*, J. Differential Geom. 53 (1999) 327–358 (where the question
+already appears). The question traces back to Gromov's essay, §4.5.C.
+
+## Work done
+
+No computation; pure reasoning. I worked out the standard reduction that localizes exactly where the
+problem is stuck, and checked it against the cited literature.
+
+**Step 1 (torsion ⟹ elliptic).** Every finite subgroup of a group acting on a complete CAT(0) space
+fixes a point: a finite orbit is bounded, and a bounded set in a complete CAT(0) space has a unique
+circumcenter (Bridson–Haefliger II.2). Hence every torsion element of $G$ is elliptic.
+
+**Step 2 (boundary dichotomy).** A finitely generated group acting on a complete CAT(0) space with
+all elements elliptic either has bounded orbits — hence a global fixed point — or fixes a point of
+the visual boundary $\partial X$ (this dichotomy is the standard one used throughout the cited
+literature, e.g. Caprace–Monod, and in finite telescoping dimension Caprace–Lytchak, Math. Ann. 346
+(2010), DOI 10.1007/s00208-009-0381-1). Since a properly discontinuous action has finite point
+stabilizers, an *infinite* torsion $G$ as in (1) cannot fix a point of $X$. Conclusion:
+
+> Any counterexample to (1) is necessarily of **parabolic type**: $G$ has unbounded orbits and fixes
+> a (unique, in the visibility case) point $\xi \in \partial X$.
+
+**Step 3 (horocyclic reduction).** For isometries fixing $\xi$, the Busemann cocycle gives a
+homomorphism $G_\xi \to \mathbb{R}$; it vanishes on torsion elements, so a torsion $G$ preserves
+every horosphere centered at $\xi$. Horoballs are closed and convex (Busemann functions are convex),
+so $G$ acts properly discontinuously on a nested family of complete CAT(0) subspaces exhausting $X$,
+all with the fixed point "at infinity". The whole difficulty of the problem is concentrated in this
+horocyclic picture; Ji–Wu note one may even reduce to a proper CAT($-1$) (hence visibility) space,
+so the remaining obstruction in (1) is precisely the *bounded packing / non-visibility* pathology of
+general CAT(0) boundaries.
+
+**Step 4 (cocompact case, part (2)).** If $G$ acts properly and cocompactly on the proper CAT(0)
+space $X$, all elements of $G$ are semisimple (Bridson–Haefliger), so torsion elements are elliptic.
+Let $H \le G$ be an infinite torsion subgroup; $H$ also acts properly discontinuously.
+- If $H$ is finitely generated, Step 2 forces $H$ to fix $\xi \in \partial X$. Papasoglu–Swenson's
+  Theorem 3.17 says $H$ cannot fix any point of $\Lambda H$; so a counterexample would need
+  $\xi \notin \Lambda H$ with $H$ acting horocyclically at a point disjoint from its limit set. In
+  visibility spaces this is contradictory (Ji–Wu exploit exactly this); in general CAT(0) spaces the
+  argument breaks — this is the precise gap.
+- If $H$ is infinitely generated and locally finite, one can go slightly further (my own elementary
+  observation, standard tools): $H$ is amenable, so by Adams–Ballmann (*Amenable isometry groups of
+  Hadamard spaces*, Math. Ann. 1998 — classical, not re-verified this session) $H$ either fixes a
+  point of $\partial X$ or preserves a Euclidean flat $F \cong \mathbb{R}^k \subset X$. In the flat
+  case, write $H = \bigcup_n F_n$ as an ascending union of finite subgroups; $\mathrm{Fix}(F_n) \cap F$
+  is a descending chain of affine subspaces of $\mathbb{R}^k$, which stabilizes for dimension
+  reasons, giving a global fixed point of $H$ in $X$ — contradicting properness ($H$ infinite, point
+  stabilizers finite). So a locally finite counterexample must *also* fix a boundary point and
+  preserve no flat: exactly the same hard case as for f.g. $H$.
+
+So the naive approaches all funnel into one unresolved configuration: a (locally finite or f.g.)
+infinite torsion group acting properly, horocyclically, fixing a boundary point of a proper CAT(0)
+space. I could not rule this out in general — this matches the literature, where the identical
+configuration is the acknowledged obstacle.
+
+## Result
+
+**Open.** Neither part is solved in general, and I did not solve them. The triage above shows the
+problem reduces to a single hard configuration (horocyclic torsion actions fixing a boundary point),
+which is resolved — negatively for the torsion group — in every setting with extra structure:
+dimension ≤ 2, cube complexes, visibility + bounded packing, subexponential growth, Helly graphs and
+classical buildings, systolic/small-cancellation complexes. No example of an infinite finitely
+generated torsion group acting properly discontinuously on *any* proper CAT(0) space (of any
+dimension) is known; conversely, infinite-dimensionality and loss of finite generation or of
+cocompactness are all known to allow torsion phenomena, so the hypotheses are sharp. The strongest
+evidence for a negative answer to (1) in finite dimensions: every natural candidate (Grigorchuk-type
+groups of intermediate growth) is now provably excluded by Izeki–Karlsson, and exponential-growth
+torsion groups (Burnside-type) fail all known structural footholds.
+
+Classification: **OPEN-TRIAGE** (parts (1) and (2) open; extensive verified partial results; the
+precise remaining gap identified).
+
+## What remains
+
+- Part (1), finite-dimensional $X$: the full NOP Conjecture 1.5 — does every f.g. torsion group
+  acting on a finite-dimensional CAT(0) space fix a point? Open already in dimension 3. The key test
+  case: do infinite Burnside groups $B(m,n)$ (large odd $n$) act properly on a proper
+  finite-dimensional CAT(0) space?
+- Part (1), infinite-dimensional proper $X$: completely open — no positive example is known either
+  (Grigorchuk/Burnside actions on infinite-dimensional cube complexes a la Osajda's cubization are
+  not properly discontinuous on proper spaces; Grigorchuk groups act properly on Hilbert space,
+  which is not proper).
+- Part (2): does a cocompact CAT(0) group contain an infinite torsion subgroup? Open even for $X$ a
+  3-dimensional CAT(0) complex; resolved for cube complexes (f.g. subgroups, via Sageev) and with
+  structural constraints by Genevois and Papasoglu–Swenson. The infinitely generated locally finite
+  case reduces (via Adams–Ballmann) to the same horocyclic obstruction.
+- Remove the bounded-packing hypothesis from Ji–Wu's visibility-space theorem, or extend
+  Izeki–Karlsson's random-walk/harmonic-map method beyond the weakly Liouville (zero-drift) class.
+- Items not independently verified against the publisher record in this session: Leder–Varghese
+  [LV20] (cited via [HO21]); Adams–Ballmann 1998 (classical); the journal publication status of
+  [HO21] and [Izeki–Karlsson] (cited as preprints); [GLU24]'s exact scope (cited via
+  [Izeki–Karlsson]); whether the wreath-product cube complexes in Genevois's example are locally
+  finite.

research/AMR-010-0212.md

@@ -0,0 +1,175 @@
+---
+id: AMR-010-0212
+classification: PARTIAL-PROGRESS
+wording_corrected: no
+---
+
+# AMR-010-0212 — Coxeter groups acting on CAT(0) spaces: convex cocompactness of special subgroups (Bestvina Q 2.12, Kim Ruane)
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is faithful to the source. The original wording, from M. Bestvina's
+problem list *Questions in Geometric Group Theory* (2004), Question 2.12, attributed to Kim Ruane
+(verified against the author-hosted PDF text, which reads "Q 2.12. (Kim Ruane) Let G be a Coxeter
+group, e.g. right-angled, and assume that G acts properly discontinuously and by isometries on a
+..." and "...Suppose that H is a special subgroup of G. Is there a closed convex subset..."):
+[Bestvina's list](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf).
+
+> Let $G$ be a Coxeter group, e.g. right-angled, and assume that $G$ acts properly discontinuously
+> and by isometries on a CAT(0) space $X$. How is $X$ different from the Coxeter complex?
+> Specifically, if $H$ is a special subgroup of $G$, is there a closed convex subset of $X$ on
+> which $H$ acts cocompactly?
+
+Here "special subgroup" means a standard parabolic subgroup $W_T = \langle T \rangle$, $T \subseteq S$,
+for the Coxeter system $(G,S)$, and "convex" is metric convexity in the CAT(0) metric.
+
+## Status / Literature
+
+**Open in general.** I found no paper that states or answers this question directly, and no
+source listing it as solved. The question sits at the intersection of several active programs;
+the surrounding literature gives both positive partial answers and warnings.
+
+Verified references (existence checked via the arXiv API, journal/publisher pages, or multiple
+independent published reference lists; Crossref's API was unreachable from this environment):
+
+1. **The Coxeter/Davis complex itself.** The Coxeter complex, and its non-positively curved
+   refinement the Davis–Moussong complex (Moussong's thesis: G. Moussong, *Hyperbolic Coxeter
+   groups*, Ph.D. thesis, Ohio State, 1988 — verified via
+   [arXiv:2511.20559 ref. Mou88](https://arxiv.org/html/2511.20559v1); see also M. Davis,
+   *The geometry and topology of Coxeter groups*, Princeton, 2008 — verified via
+   [arXiv:1703.09032 ref. Dav08](https://arxiv.org/pdf/1703.09032)), is CAT(0), and every special
+   subgroup $W_T$ acts cocompactly on the subcomplex of cosets of $W_T$, which is closed and
+   convex. So for $X$ = Coxeter/Davis complex the answer is trivially yes; the question is whether
+   this persists for *arbitrary* proper CAT(0) actions.
+
+2. **Cubulations of Coxeter groups.** G. A. Niblo and L. D. Reeves, *Coxeter groups act on CAT(0)
+   cube complexes*, J. Group Theory 6 (2003), no. 3, 399–413, MR1983376 — verified via
+   [arXiv:2511.20559 ref. NR03](https://arxiv.org/html/2511.20559v1). F. Haglund and D. T. Wise,
+   *Coxeter groups are virtually special*, Adv. Math. 224 (2010), 1890–1903 — verified via
+   [arXiv:1501.07001 ref. 24](https://arxiv.org/pdf/1501.07001); and *Special cube complexes*,
+   Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620, MR2377497 — verified via
+   [arXiv:2511.20559 ref. HW08](https://arxiv.org/html/2511.20559v1). The Niblo–Reeves cubulation
+   is proper but is cocompact only when $W$ has no irreducible affine parabolic subgroups of
+   rank $\ge 3$ (as recalled in [FLS] below, citing Williams and Caprace–Mühlherr), so the
+   "properly discontinuous" hypothesis of the question is genuinely weaker than geometricity.
+
+3. **Convex-cocompactness across all cubulations (most relevant recent work).**
+   E. Fioravanti, I. Levcovitz, M. Sageev, *Coarse cubical rigidity*, J. Topol. 17 (2024), no. 3,
+   e12353; arXiv:2210.11418, DOI 10.1112/topo.12353 — verified via the
+   [arXiv API](http://export.arxiv.org/api/query?id_list=2210.11418) and read in detail via
+   [ar5iv](https://ar5iv.labs.arxiv.org/html/2210.11418). They show that two cocompact cubulations
+   of a group induce the same coarse median structure iff they have the same convex-cocompact
+   subgroups, and prove (their Theorem B): for a right-angled Coxeter group $W_\Gamma$, a cocompact
+   cubulation $X$ has the same convex-cocompact subgroups as the Davis complex — hence **all
+   special subgroups are convex-cocompact in $X$** — provided either (1) the action is strongly
+   cellular, or (2) every infinite dihedral special subgroup $\langle x,y\rangle$ ($xy$ of infinite
+   order) is convex-cocompact in $X$. Their Corollary C: if $\Gamma$ has no "loose squares", then
+   $W_\Gamma$ has a *unique* cubical coarse median structure, so in **every** cocompact cubulation
+   all special subgroups are convex-cocompact (combinatorial sense: invariant convex subcomplex).
+   Conversely, for graphs with loose squares they construct exotic cubulations (their Example 5.5)
+   with non-standard coarse median structure, where by the contrapositive of Theorem B(2) some
+   infinite dihedral special subgroup is *not* combinatorially convex-cocompact. The general
+   quasiconvex-vs-convex issue for cube complexes is the subject of M. Sageev and D. T. Wise,
+   *Cores for quasiconvex actions*, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2731–2741 —
+   verified via [arXiv:1908.09046 ref. SW15](https://arxiv.org/pdf/1908.09046) and the
+   [AMS journal listing](https://documat.unirioja.es/ejemplar/400691).
+
+4. **How different can $X$ be?** C. B. Croke and B. Kleiner, *Spaces with nonpositive curvature
+   and their ideal boundaries*, Topology 39 (2000), no. 3, 549–556, MR1746908 — verified via
+   [arXiv:2603.23141 ref. 11](https://arxiv.org/html/2603.23141v1) and
+   [arXiv:2603.05742 ref. 3](https://arxiv.org/html/2603.05742v2): a single CAT(0) group can act
+   geometrically on two CAT(0) spaces with non-homeomorphic visual boundaries. So the answer to the
+   first sentence of Ruane's question ("how is $X$ different from the Coxeter complex?") is:
+   potentially very different, at least at the level of the visual boundary.
+
+5. Reflection-rigidity context: P.-E. Caprace and B. Mühlherr, *Conjugacy of 2-spherical subgroups
+   of Coxeter groups and parallel walls*, Algebr. Geom. Topol. 6 (2006), 1987–2029 — verified via
+   the [MSP journal page](https://msp.org/agt/2006/6-4/agt-v6-n4-p15-p.pdf): for infinite
+   irreducible 2-spherical Coxeter groups the Coxeter generating set (hence the notion of special
+   subgroup) is intrinsic up to diagram twists, so the question does not depend on a choice of
+   generating set in those cases.
+
+## Work done
+
+**Reduction (own derivation, folklore-level).** Special subgroups are undistorted in $G$: by the
+deletion condition for Coxeter systems, every reduced $S$-word for an element $w \in W_T$ uses
+only letters of $T$, so the Cayley graph of $(W_T,T)$ embeds isometrically (as a full convex
+subgraph) in the Cayley graph of $(G,S)$. Hence for every geometric (proper + cocompact) action of
+$G$ on a CAT(0) space $X$, the orbit map shows every $W_T$-orbit is quasi-isometrically embedded,
+i.e. $W_T$ is a *quasiconvex* subgroup for the action. Ruane's specific question is therefore
+equivalent (for geometric actions) to:
+
+> Is every quasiconvex special subgroup of a Coxeter group convex-cocompact in every CAT(0) space
+> on which the group acts geometrically?
+
+This is the CAT(0) instance of the general quasiconvex-vs-convex gap (quasiconvex subgroups of
+CAT(0) groups need not act cocompactly on convex hulls in general).
+
+**Positive answer in the hyperbolic case (own derivation, certainly folklore).** Suppose $G$ is
+word-hyperbolic and acts geometrically on a CAT(0) space $X$. Then $X$ is quasi-isometric to $G$
+(Švarc–Milnor), hence $\delta$-hyperbolic, since hyperbolicity is a quasi-isometry invariant of
+geodesic spaces (see Bridson–Haefliger, *Metric spaces of non-positive curvature*, Springer 1999 —
+verified via [arXiv:2603.05742 ref. 2](https://arxiv.org/html/2603.05742v2)). Let $H=W_T$ and fix
+$x_0 \in X$. The orbit $H x_0$ is $Q$-quasiconvex by the reduction above. In a $\delta$-hyperbolic
+geodesic space, the convex hull of a $Q$-quasiconvex subset lies in its $R(Q,\delta)$-neighbourhood
+(any point of the hull lies on a geodesic between orbit points, which lies near the orbit).
+Hence $C := \mathrm{Conv}(H x_0)$ is a closed, convex, $H$-invariant subset of $X$ contained in
+the $R$-neighbourhood of $H x_0$; since $H$ acts properly on $X$ and cocompactly on its own
+orbit's neighbourhood, $H \backslash C$ is compact. So the answer to Ruane's question is **yes**
+whenever the Coxeter group is word-hyperbolic (e.g. all hyperbolic right-angled Coxeter groups).
+Consistently, Haglund's theorem (cited as [Hag08, Theorem H] inside the verified [FLS] paper)
+says that for hyperbolic groups, convex-cocompact subgroups in any cocompact cubulation are
+exactly the quasiconvex ones.
+
+**Analysis of the cubical obstruction (own synthesis of [FLS]).** The most serious known
+obstruction comes from Fioravanti–Levcovitz–Sageev: for RACGs $W_\Gamma$ with loose squares there
+exist cocompact cubulations in which some infinite dihedral special subgroup fails to stabilize any
+convex *subcomplex* cocompactly. However, this does **not** answer Ruane's literal question,
+because she allows arbitrary closed *metrically* convex subsets, not only subcomplexes, and the two
+notions differ. Witness (their own running example, which I checked in detail): the standard
+square tiling of $\mathbb{R}^2$ with the $\pi/4$-rotated action of
+$D_\infty \times D_\infty$ — each factor preserves a diagonal line, which is a closed CAT(0)-convex
+subset on which that special subgroup acts cocompactly, although the only convex *subcomplex*
+containing it is the whole plane. Thus a special subgroup can fail combinatorial
+convex-cocompactness while still acting cocompactly on a closed convex subset. Whether the exotic
+[FLS] cubulations (Example 5.5 there) can be upgraded to violate the metric version is, as far as
+I could determine, not addressed in the literature.
+
+**Triage.** Searched for any work citing Bestvina's Q 2.12 or addressing "Coxeter special subgroup
+convex cocompact CAT(0)" directly; the question is not resolved anywhere I could find. The closest
+systematic framework is coarse cubical/median rigidity [FLS]; the first part of the question
+("how is $X$ different from the Coxeter complex?") is matched by the boundary non-uniqueness
+phenomenon of Croke–Kleiner.
+
+## Result
+
+- **Verified the source and wording**: Bestvina, *Questions in Geometric Group Theory*, Q 2.12
+  (Kim Ruane); the dataset transcription is accurate, no correction needed.
+- **Established the status: open**, with a precise reduction: for geometric actions the question is
+  equivalent to "quasiconvex $\Rightarrow$ convex-cocompact" restricted to special subgroups of
+  Coxeter groups.
+- **Proved (own work, folklore-level) the positive answer when $G$ is word-hyperbolic**: $W_T$
+  acts cocompactly on the convex hull of any orbit, which is closed and convex.
+- **Documented strong positive partial results** in the cubical category (Fioravanti–Levcovitz–
+  Sageev 2024): all special subgroups are convex-cocompact in every cocompact cubulation of a RACG
+  whose defining graph has no loose squares, and in every strongly cellular cocompact cubulation.
+- **Identified the precise gap**: known exotic cubulations only obstruct *combinatorial*
+  convex-cocompactness (convex subcomplexes), while Ruane's question asks for closed *metrically*
+  convex subsets; the rotated $D_\infty\times D_\infty$ example shows the two genuinely differ, so
+  even in the cubical case her literal question is not settled by [FLS].
+
+## What remains
+
+- The general question for a geometric action on an arbitrary (non-hyperbolic, non-cubical) CAT(0)
+  space: is every special subgroup convex-cocompact in the metric sense?
+- Even in the cubical setting: for RACGs with loose squares, do the FLS exotic cubulations admit a
+  special subgroup that fails to act cocompactly on *every* closed metrically convex subset? A
+  negative answer there would refute Ruane's question in its strongest reading; a positive answer
+  would need a metric (not combinatorial) core theorem for parabolics.
+- The proper-but-not-cocompact regime the question explicitly allows (e.g. non-cocompact
+  Niblo–Reeves cubulations of Coxeter groups with rank $\ge 3$ irreducible affine parabolics) is
+  essentially untouched.
+- Note on verification: all citations above were checked against the arXiv API, journal/publisher
+  pages, or multiple independent published reference lists; Crossref's API was unreachable from
+  this environment, so DOI-level confirmation was not possible for the older journal items (their
+  bibliographic data agree across all independent sources checked).

research/AMR-010-0213.md

@@ -0,0 +1,150 @@
+---
+id: AMR-010-0213
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0213 — Classify Coxeter groups up to isomorphism
+
+## Problem (corrected statement if needed)
+
+The worklist transcription matches the source verbatim. From M. Bestvina,
+*Questions in Geometric Group Theory* (updated July 2004), Question 2.13
+(p. 9), <https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>:
+
+> **Q 2.13. (Ruth Charney)** Classify Coxeter groups up to isomorphism.
+
+Bestvina's own remark: "Interesting examples of isomorphic Coxeter groups (and
+Artin groups) with non-isomorphic diagrams were given in [BMMN02]. In the
+opposite direction, conditions under which isomorphism of groups implies
+isomorphism of diagrams were given in [CD00], [Rad03], ..."
+
+The standard precise formulation (e.g. Caprace–Mühlherr, Oberwolfach 2004):
+determine all pairs of Coxeter systems (W₁, S₁), (W₂, S₂) such that the
+abstract groups W₁ and W₂ are isomorphic; equivalently, given an abstract
+Coxeter group W, determine all subsets S ⊂ W such that (W, S) is a Coxeter
+system.
+
+## Status / Literature
+
+**Open in general** — this is the classical *Isomorphism Problem for Coxeter
+Groups*, still unsolved as of 2026. Recent confirmations: T. Gobet's HDR
+thesis (2023, <https://gobet.perso.math.cnrs.fr/hdr.pdf>) states "in general
+the question is still open, and known as 'The Isomorphism Problem for Coxeter
+Groups'"; Google DeepMind's formal-conjectures tracker lists it as a
+"classical open problem" (issue #2147, Feb 2026). All citations below were
+verified against Crossref or the arXiv API.
+
+*Non-rigidity (the problem is nontrivial):*
+- N. Brady, J. McCammond, B. Mühlherr, W. Neumann, *Rigidity of Coxeter
+  groups and Artin groups*, Geom. Dedicata 94 (2002), 91–109,
+  doi:10.1023/A:1020948811381 — isomorphic Coxeter groups (and Artin groups)
+  with non-isomorphic diagrams, produced by "twists" along separating edges.
+  (Verified via Crossref; this is Bestvina's [BMMN02].)
+- Already among finite groups: I₂(2m) ≅ A₁ × I₂(m) for m odd, e.g.
+  I₂(6) ≅ A₁ × I₂(3) (dihedral of order 12 ≅ Z/2 × S₃), so distinct Coxeter
+  systems give the same abstract group. The abstract-isomorphism
+  classification of finite Coxeter groups is known (B. Mühlherr, *On
+  isomorphisms between Coxeter groups*, Des. Codes Cryptogr. 21 (2000);
+  bibliographic data seen only in the Crossref-verified reference list of
+  BMMN02, not independently verified).
+
+*Rigidity results (isomorphic groups ⇒ isomorphic/strongly equivalent systems):*
+- D. G. Radcliffe, *Rigidity of right-angled Coxeter groups*,
+  arXiv:math/9901049 (verified via arXiv API): any two Coxeter generating
+  sets of a right-angled Coxeter group are conjugate (strong rigidity);
+  hence RACGs are classified up to isomorphism by their defining graphs.
+- D. G. Radcliffe, *Rigidity of graph products of groups*, Algebr. Geom.
+  Topol. 3 (2003), 1079–1088, doi:10.2140/agt.2003.3.1079 (verified via
+  Crossref; this is Bestvina's [Rad03]).
+- R. Charney, M. Davis, *When is a Coxeter system determined by its Coxeter
+  group?*, J. London Math. Soc. (2) 61 (2000), 441–461,
+  doi:10.1112/S0024610799008583 (verified via Crossref; Bestvina's [CD00]).
+- B. Mühlherr, *Automorphisms of graph-universal Coxeter groups*, J. Algebra
+  200 (1998), 629–649, doi:10.1006/jabr.1997.7230 (verified via Crossref
+  reference data) — rigidity for graph-universal (all m(s,t) ∈ {2,∞}) groups.
+- B. Mühlherr, R. Weidmann, *Rigidity of skew-angled Coxeter groups*, Adv.
+  Geom. 2 (2002), 391–415, doi:10.1515/advg.2002.018 (verified via Crossref
+  reference data).
+- P. Bahls, *A new class of rigid Coxeter groups*, Internat. J. Algebra
+  Comput. 13 (2003), 87–94, doi:10.1142/S0218196703001377 (verified via
+  Crossref); and his monograph *The Isomorphism Problem in Coxeter Groups*,
+  Imperial College Press, 2005, doi:10.1142/9781860947193 (verified).
+- P.-E. Caprace, B. Mühlherr, *Reflection rigidity of 2-spherical Coxeter
+  groups*, Proc. London Math. Soc. 94 (2007), 520–542,
+  doi:10.1112/plms/pdl015 (verified via Crossref).
+
+*Structural reduction:*
+- R. Howlett and B. Mühlherr (~2004, unpublished preprint) reduced the
+  general isomorphism problem to its *reflection-preserving* version
+  (isomorphisms sending reflections to reflections). Surveyed in
+  B. Mühlherr, *The isomorphism problem for Coxeter groups*,
+  arXiv:math/0506572 (verified via arXiv API), published in *The Coxeter
+  Legacy*, Fields Inst. Comm., 2006. The operative conjectural answer is that
+  (up to the known exceptions) isomorphic Coxeter systems differ by diagram
+  twists ("twist equivalence").
+
+*Recent related direction:* S. André, G. Paolini, *Around first-order
+rigidity of Coxeter groups*, arXiv:2407.01164 (2024; seen via arXiv listing)
+— Coxeter groups with spherical/affine/hyperbolic irreducible components are
+first-order rigid among finitely torsion-generated groups; orthogonal to but
+motivated by the isomorphism problem.
+
+## Work done
+
+1. Located and read the source: the worklist text is an exact transcription
+   of Q 2.13 of Bestvina's list; no wording correction needed.
+2. Verified every cited item against Crossref/arXiv (see flags above; the two
+   items marked "reference data only" were confirmed as entries in Crossref
+   metadata of verified papers but not fetched directly).
+3. Warm-up analysis (elementary, own reasoning) of the smallest cases, which
+   already exhibits both phenomena driving the general problem:
+   - **Rank 1–2.** A rank-2 Coxeter group is I₂(m) (order 2m) or the
+     infinite dihedral group D∞ (m = ∞). Invariants separate them:
+     abelianization I₂(m)ᵃᵇ is (Z/2)² for m even and Z/2 for m odd (and
+     (Z/2)² for D∞); the center is Z/2 for m even, trivial for m odd, Z/2
+     for D∞... but as *abstract* groups collisions occur across ranks:
+     for m odd, writing r = st (order 2m) and z = r^m (central involution),
+     I₂(2m) = ⟨z⟩ × ⟨s, r²⟩ ≅ A₁ × I₂(m), because r² has odd order m and
+     ⟨s, r²⟩ is dihedral of order 2m. Conversely these are the *only*
+     collisions among direct products of rank ≤ 2 systems: order, |Wᵃᵇ|,
+     and the center distinguish all remaining pairs. Hence even the
+     rank-2 abstract classification forces one to allow decompositions
+     into different numbers of irreducible factors.
+   - The two general obstruction mechanisms match this picture:
+     (a) non-reflection-preserving isomorphisms (as above, reflections of
+     one system are not reflections in the other) — controlled in principle
+     by the Howlett–Mühlherr reduction; (b) reflection-preserving
+     isomorphisms that are not diagram isomorphisms, conjecturally all
+     explained by twists (BMMN02-type), settled positively only in special
+     classes (skew-angled, graph-universal, 2-spherical cases above).
+
+## Result
+
+**OPEN-TRIAGE.** The problem — classifying Coxeter groups up to abstract
+group isomorphism — is open in full generality and I did not solve it (no
+serious attempt is feasible: it is a flagship open problem of the area).
+What is established: the problem reduces (Howlett–Mühlherr) to the
+reflection-preserving isomorphism problem; large classes are (strongly)
+rigid, notably right-angled Coxeter groups, where the classification reduces
+to graph isomorphism (Radcliffe); graph-universal, skew-angled, new classes
+of Bahls, and reflection-rigid 2-spherical groups (Mühlherr, Mühlherr–
+Weidmann, Bahls, Caprace–Mühlherr); non-rigid examples arise from diagram
+twists (Brady–McCammond–Mühlherr–Neumann) and from decomposability
+phenomena already visible in rank 2. The conjectural complete answer is
+"twist equivalence up to the known finite/decomposable exceptions."
+
+## What remains
+
+- The general reflection-preserving isomorphism problem, especially for
+  Coxeter systems that are neither 2-spherical nor in the known rigid
+  classes; proof (or counterexamples) of the twist-equivalence conjecture.
+- Publication/completion of the Howlett–Mühlherr reduction (still
+  unpublished as far as I could verify).
+- Algorithmic aspect: no general algorithm is known that decides, given two
+  Coxeter diagrams, whether the groups are abstractly isomorphic (known for
+  right-angled groups via Radcliffe + graph isomorphism).
+- Honesty note: I did not verify the current status of the twist-equivalence
+  conjecture beyond the sources above; a deeper 2015–2026 literature sweep
+  (e.g. work of Nuida, Marquis, Grant–Levcovitz on reflection rigidity) was
+  beyond the fetch budget.

research/AMR-010-0214.md

@@ -0,0 +1,194 @@
+---
+id: AMR-010-0214
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0214 — Classify Artin groups up to isomorphism
+
+## Problem (corrected statement)
+
+The dataset transcription matches the source exactly. Question 2.14 of M. Bestvina's
+problem list *Questions in Geometric Group Theory* (author-hosted PDF,
+[questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), p. 9):
+
+> **Q 2.14 (Ruth Charney).** Classify Artin groups up to isomorphism.
+
+Precise formulation. Let $\Gamma$ be a finite simplicial graph with vertex set $S$ and
+edges $\{s,t\}$ labeled by integers $m_{st}\in\{2,3,\dots,\infty\}$ ($m_{st}=\infty$
+means "no edge / no relation"). The *Artin group* of $\Gamma$ is
+
+$$A_\Gamma=\Big\langle S \;\Big|\; \underbrace{sts\cdots}_{m_{st}}=\underbrace{tst\cdots}_{m_{st}}\ \text{for each edge } \{s,t\}\Big\rangle .$$
+
+The problem asks for a complete classification of the groups $A_\Gamma$ up to abstract
+group isomorphism — equivalently, a decision procedure (or a complete, computable set of
+invariants) telling when $A_\Gamma\cong A_{\Gamma'}$ for non-isomorphic defining graphs.
+It subsumes the **isomorphism problem for Artin groups**: is there an algorithm that,
+given $\Gamma,\Gamma'$, decides whether $A_\Gamma\cong A_{\Gamma'}$?
+
+## Status / Literature
+
+**Open in general** (as of August 2026). The problem is solved within several natural
+classes, and there is an active conjectural framework (the *twist conjecture*) that
+would settle it completely. All references below were verified against Crossref and/or
+the arXiv API on 2026-08-04.
+
+Solved classes:
+
+- **Right-angled Artin groups** (all $m_{st}=2$): C. Droms,
+  *Isomorphisms of graph groups*, Proc. Amer. Math. Soc. 100 (1987), 407–408,
+  [DOI 10.1090/S0002-9939-1987-0891135-1](https://doi.org/10.1090/s0002-9939-1987-0891135-1):
+  $A_\Gamma\cong A_{\Gamma'}$ iff $\Gamma\cong\Gamma'$ as graphs.
+- **Spherical-type Artin groups** (associated Coxeter group finite): L. Paris,
+  *Artin groups of spherical type up to isomorphism*, J. Algebra 281 (2004), 666–678,
+  [DOI 10.1016/j.jalgebra.2004.04.021](https://doi.org/10.1016/j.jalgebra.2004.04.021),
+  solves the isomorphism problem within this class.
+- **Large-type Artin groups** (all $m_{st}\ge 3$): N. Vaskou,
+  *The isomorphism problem for large-type Artin groups*,
+  [arXiv:2201.08329](https://arxiv.org/abs/2201.08329) (v3, 2023): two large-type
+  Artin groups are isomorphic iff their defining graphs are *twist equivalent*
+  (see below). The abstract states this "answers several questions raised by Charney"
+  — i.e., directly addresses the present problem within this class.
+  The companion paper *Automorphisms of large-type free-of-infinity Artin groups*,
+  Geom. Dedicata 219 (2025), art. 16,
+  [DOI 10.1007/s10711-024-00951-x](https://doi.org/10.1007/s10711-024-00951-x),
+  computes the automorphism groups.
+
+The twist conjecture framework:
+
+- N. Brady, J. McCammond, B. Mühlherr, W. Neumann,
+  *Rigidity of Coxeter Groups and Artin Groups*, Geom. Dedicata 94 (2002), 91–109,
+  [DOI 10.1023/A:1020948811381](https://doi.org/10.1023/A:1020948811381),
+  introduced diagram mutations/twists. An *elementary twist* of $\Gamma$ along a
+  separating subset $Y\subseteq S$ spanning an indecomposable spherical-type subdiagram
+  conjugates one component of $\Gamma-\Gamma_Y$ by the Garside element $\Delta_Y$;
+  twist-equivalent graphs therefore yield isomorphic Artin groups (the easy direction).
+  The **twist conjecture for Artin groups** asserts the converse:
+  $A_\Gamma\cong A_{\Gamma'}$ iff $\Gamma$ and $\Gamma'$ are twist equivalent.
+  Since twist equivalence of finite labeled graphs is decidable by finite search, the
+  conjecture would give a complete solution of Q 2.14, including decidability.
+- J. Crisp, *Automorphisms and abstract commensurators of 2-dimensional Artin groups*,
+  Geom. Topol. 9 (2005), 1381–1441,
+  [DOI 10.2140/gt.2005.9.1381](https://doi.org/10.2140/gt.2005.9.1381):
+  computed the isomorphism groupoid (all isomorphisms between standard parabolics) for
+  connected large-type triangle-free Artin groups.
+- A. Martin, N. Vaskou, *Characterising large-type Artin groups*, Bull. London Math.
+  Soc. 56 (2024), 3346–3357,
+  [DOI 10.1112/blms.13136](https://doi.org/10.1112/blms.13136): the *class* of
+  large-type Artin groups is invariant under isomorphism (an Artin group isomorphic to a
+  large-type one is itself large-type), and they describe all Artin groups isomorphic
+  to a given large-type one.
+- O. Jones, G. Mangioni, G. Sartori,
+  *A combination theorem for the twist conjecture for Artin groups*, J. Algebra 701
+  (2026), 436–476, [arXiv:2507.13971](https://arxiv.org/abs/2507.13971),
+  [DOI 10.1016/j.jalgebra.2026.04.016](https://doi.org/10.1016/j.jalgebra.2026.04.016):
+  reduces a strong form of the twist conjecture to the case of defining graphs **without
+  separating vertices**, and produces new classes satisfying the conjecture.
+- Very recent: *Isomorphism invariance of the girth of Artin groups*
+  ([arXiv:2601.05078](https://arxiv.org/abs/2601.05078), 2026 preprint) — further
+  isomorphism invariants of the defining graph (existence verified via arXiv listing;
+  contents not checked in detail). An AIM workshop report
+  ([aimath.org/pastworkshops/geomartingprep.pdf](https://aimath.org/pastworkshops/geomartingprep.pdf))
+  records ongoing work extending Vaskou's methods from large-type to all
+  2-dimensional Artin groups, with a proof sketch that even-type 2-dimensional Artin
+  groups are determined by their presentation graphs.
+
+Classical background used below: E. Brieskorn, K. Saito, *Artin-Gruppen und
+Coxeter-Gruppen*, Invent. Math. 17 (1972), 245–271,
+[DOI 10.1007/BF01406235](https://doi.org/10.1007/BF01406235) (Garside structure,
+centers of spherical-type Artin groups).
+
+## Work done
+
+1. **Verified the wording** against the source: the search-indexed text of Bestvina's
+   PDF reads "Q 2.14. (Ruth Charney) Classify Artin groups up to isomorphism." — the
+   dataset transcription is exact, so `wording_corrected: no`.
+2. **Verified every citation** above via Crossref (`api.crossref.org/works?...`) or the
+   arXiv API (`export.arxiv.org/api/query?id_list=...`); DOIs, volumes, pages and dates
+   are as returned by those services.
+3. **A self-contained rigorous solution of the 2-generator (dihedral) case**, derived
+   here directly (it is of course subsumed by Paris 2004, since $I_2(m)$ is spherical,
+   and by Vaskou for $m\ge 3$):
+
+**Theorem (dihedral case).** Let
+$DA_m=\langle s,t \mid \mathrm{prod}(s,t;m)=\mathrm{prod}(t,s;m)\rangle$, $m\ge 2$.
+Then $DA_m\cong DA_n$ iff $m=n$.
+
+*Proof.* Write $u=st$.
+
+- $m=2$: the relation is $st=ts$, so $DA_2=\mathbb Z^2$ is abelian. For $m\ge 3$,
+  $DA_m$ is nonabelian (it surjects onto the nonabelian Coxeter group $W(I_2(m))$), so
+  $DA_2$ is distinguished from all $DA_m$, $m\ge3$.
+- For $m\ge3$ the center of $DA_m$ is infinite cyclic, generated by the Garside element
+  $\Delta=\mathrm{prod}(s,t;m)$ if $m$ is even and by $\Delta^2=u^m$ if $m$ is odd
+  (Brieskorn–Saito 1972). Note $Z(DA_m)=\langle u^{m/2}\rangle$ ($m$ even) resp.
+  $\langle u^m\rangle$ ($m$ odd). The center is intrinsically defined, hence the central
+  quotient $Q_m=DA_m/Z(DA_m)$ is an isomorphism invariant.
+- **$m$ even.** In $Q_m$, $u^{m/2}=1$, and then the braid relation holds automatically:
+  $(ts)^{m/2}=s^{-1}(st)^{m/2}s=1$. Hence
+  $Q_m=\langle s,t\mid (st)^{m/2}=1\rangle=\langle s\rangle * \langle st\rangle
+  \cong \mathbb Z * \mathbb Z_{m/2}$.
+  In a free product $\mathbb Z*\mathbb Z_k$ every finite-order element is conjugate into
+  $\mathbb Z_k$, so the maximal torsion order is exactly $k=m/2$, and $m$ is recovered.
+- **$m$ odd.** Write $k=(m-1)/2$. The braid relation is $u^ks=(ts)^kt=s^{-1}u^kst$,
+  i.e. $su^ks=u^{k+1}$. Since $k+(k+1)=m$, imposing $u^m=1$ turns this into
+  $(su^k)^2=u^m=1$. Setting $a=su^k$ we have $s=au^{-k}$, and the presentation becomes
+  $Q_m=\langle a,u\mid a^2=u^m=1\rangle\cong \mathbb Z_2*\mathbb Z_m$.
+  Finite-order elements of $\mathbb Z_2*\mathbb Z_m$ have order $2$ or a divisor of $m$;
+  as $m\ge3$ is odd, the maximal torsion order is exactly $m$, and $m$ is recovered.
+- **Even vs. odd.** If $m$ is even and $n$ is odd, the abelianizations differ:
+  for even $m$ the relator abelianizes trivially, so $DA_m^{\mathrm{ab}}=\mathbb Z^2$;
+  for odd $m$ it forces $s=t$, so $DA_n^{\mathrm{ab}}=\mathbb Z$. Hence
+  $DA_m\not\cong DA_n$.
+
+All cases together give $DA_m\cong DA_n\Rightarrow m=n$; the converse is trivial. $\square$
+
+4. **Triage of the general case** (why it is hard and what a solution must look like):
+   - *Easy invariants exist but are far from complete.* The abelianization is
+     $\mathbb Z^{c(\Gamma)}$, where $c(\Gamma)$ is the number of connected components of
+     the odd-labeled subgraph of $\Gamma$; the center detects irreducible spherical-type
+     parabolics; cohomological dimension is computable where the $K(\pi,1)$-conjecture
+     is known (spherical type by Deligne; 2-dimensional and FC-type by
+     Charney–Davis). None of these distinguishes twist-equivalent from
+     non-twist-equivalent graphs.
+   - *The difficulty is reconstructing the Coxeter/Deligne geometry purely
+     algebraically.* Vaskou's solution in large type proceeds by characterizing standard
+     parabolic subgroups group-theoretically and rebuilding the Deligne complex from the
+     abstract group; the Martin–Vaskou class-invariance result is what allows one to
+     conclude that no isomorphism crosses class boundaries. Extending this to graphs
+     with $2$-labels (RAAG-type parabolics, where automorphism groups are much wilder)
+     is the main open front; the Jones–Mangioni–Sartori combination theorem reduces the
+     problem to graphs with no separating vertices, where such parabolics cannot be
+     "twisted apart", but the indivisible case with $2$-labels remains unresolved.
+
+## Result
+
+The general problem is **open**. What is established:
+
+- The classification is completely solved within the classes of right-angled (Droms
+  1987), spherical-type (Paris 2004), and large-type (Vaskou 2023) Artin groups, and
+  the large-type class is isomorphism-invariant (Martin–Vaskou 2024).
+- A conjectural complete answer exists — the twist conjecture:
+  $A_\Gamma\cong A_{\Gamma'}\iff\Gamma,\Gamma'$ twist equivalent — proved in all the
+  solved cases above, and reduced to defining graphs without separating vertices
+  (Jones–Mangioni–Sartori 2026). It would also imply decidability of the isomorphism
+  problem.
+- Derived independently in this report: the dihedral classification
+  $DA_m\cong DA_n\iff m=n$, via the intrinsic central quotient
+  $DA_m/Z\cong\mathbb Z*\mathbb Z_{m/2}$ ($m$ even) resp. $\mathbb Z_2*\mathbb Z_m$
+  ($m$ odd) and abelianizations (elementary; consistent with the published results).
+
+## What remains
+
+- Prove or disprove the twist conjecture for graphs without separating vertices
+  (by the 2026 combination theorem this would settle the general case).
+- Complete the 2-dimensional case (all $m_{st}\ge 2$, dimension $\le 2$): the AIM
+  group reports a sketch for even-type 2-dimensional Artin groups; the general
+  2-dimensional case, mixing $2$-labels with higher labels, is open.
+- Show isomorphism-invariance of the remaining standard classes (e.g. that no
+  spherical-type Artin group is isomorphic to a non-spherical one, FC-type vs.
+  non-FC-type, etc.); without this, class-by-class solutions do not glue into a global
+  classification.
+- Decidability in general: even independently of the twist conjecture, no algorithm is
+  known that decides $A_\Gamma\cong A_{\Gamma'}$, and (as with Coxeter groups) none is
+  known not to exist.

research/AMR-010-0216.md

@@ -0,0 +1,181 @@
+---
+id: AMR-010-0216
+classification: OPEN-TRIAGE
+wording_corrected: no
+---
+
+# AMR-010-0216 — Are all (finite type) Artin groups CAT(0)?
+
+## Problem (corrected statement if needed)
+
+The dataset transcription is **faithful** to the source. Bestvina's "Questions in
+Geometric Group Theory" (updated July 2004), Q 2.16, reads verbatim:
+
+> **Q 2.16.** (Ruth Charney) Are all [finite type] Artin groups CAT(0)?
+> The answer is yes for small numbers of generators by the work of Krammer,
+> Tom Brady, Jon McCammond, Robert Bell. The question is open even for braid
+> groups…
+
+(Source: [questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf),
+fetched and checked line by line.) The square brackets around "finite type" are in
+Bestvina's original, so `wording_corrected: no`.
+
+Precise modern statement. An *Artin group* is given by a finite labelled Coxeter
+graph $\Gamma$: generators $S=\{\text{vertices}\}$, one relation
+$\underbrace{sts\cdots}_{m_{st}}=\underbrace{tst\cdots}_{m_{st}}$ per edge labelled
+$m_{st}\ge 2$ (no edge: $m_{st}=2$, commuting). It is of *finite (spherical) type*
+if the associated Coxeter group is finite; these are classified as
+$A_n,B_n,D_n,E_6,E_7,E_8,F_4,H_3,H_4,I_2(m)$ and products. A group is *CAT(0)* if
+it admits a geometric (proper, cocompact, by isometries) action on a CAT(0) space.
+The question — attributed to Charney (it also appears in her AIM problem list
+"Problems related to Artin groups") — asks whether every Artin group is CAT(0),
+with the finite-type case (bracketed) already open; as Bestvina notes, it was open
+in 2004 even for braid groups, and it remains so today.
+
+## Status / Literature
+
+**Open** as of August 2026, both in general and restricted to finite type; open even
+for the braid group on $n\ge 8$ strands (type $A_{n-1}$). Every citation below was
+verified against Crossref or the arXiv API.
+
+**Known CAT(0) classes.**
+
+- *Right-angled Artin groups* ($m_{st}=2$ or $\infty$): fundamental groups of
+  nonpositively curved Salvetti cube complexes (classical; see Charney–Davis,
+  "Finite $K(\pi,1)$s for Artin groups", *Prospects in Topology*, Ann. of Math.
+  Stud. 138, 1995, 110–124 — verified via Crossref reference data of
+  [10.5802/aif.3524](https://doi.org/10.5802/aif.3524)).
+- *FC type* (every clique spans a spherical parabolic): the Deligne complex with
+  the cubical/Moussong metric is CAT(0). Charney–Davis, "The $K(\pi,1)$-problem
+  for hyperplane complements associated to infinite reflection groups",
+  *J. Amer. Math. Soc.* **8** (1995), 597–627,
+  [DOI 10.2307/2152924](https://doi.org/10.2307/2152924) — verified on Crossref.
+- *2-dimensional Artin groups* (spherical parabolics of rank $\le 2$): all CAT(0);
+  some require a 3-dimensional CAT(0) space. Brady–Crisp, "Two-dimensional Artin
+  groups with CAT(0) dimension three", *Geom. Dedicata* **94** (2002), 185–214,
+  DOI 10.1023/A:1020962804856 — verified via Crossref reference data. (Caveat:
+  the Moussong metric on the 2-dimensional Deligne complex itself is *not* CAT(0)
+  in general — otherwise these groups would have CAT(0) dimension 2.)
+- *XXL type* (all labels $\ge 5$): Haettel, "XXL type Artin groups are CAT(0) and
+  acylindrically hyperbolic", *Ann. Inst. Fourier* **72** (2022), 2541–2555,
+  [DOI 10.5802/aif.3524](https://doi.org/10.5802/aif.3524) — verified on Crossref.
+- *3-generator finite type*: Brady, "Artin groups of finite type with three
+  generators", *Michigan Math. J.* **47** (2000), 313–324,
+  [DOI 10.1307/mmj/1030132536](https://doi.org/10.1307/mmj/1030132536) — verified
+  on Crossref. This settles $A_3,B_3,H_3$ and reducible rank-3 types.
+- *3-generator large type* are biautomatic: Brady–McCammond, *J. Pure Appl.
+  Algebra* **151** (2000), 1–9, DOI 10.1016/S0022-4049(99)00094-8 — verified via
+  Crossref reference data (biautomaticity is weaker evidence, not CAT(0)).
+- *3-dimensional FC type*: Bell, "Three-dimensional FC Artin groups are CAT(0)",
+  *Geom. Dedicata* **113** (2005), 21–53, DOI 10.1007/s10711-005-3691-9 —
+  verified via Crossref reference data.
+- *Braid groups (type $A_{n-1}$)*: CAT(0) for $n\le 6$ strands — Haettel–Kielak–
+  Schwer, "The 6-strand braid group is CAT(0)", *Geom. Dedicata* **182** (2016),
+  263–286, [DOI 10.1007/s10711-015-0138-9](https://doi.org/10.1007/s10711-015-0138-9)
+  — verified via arXiv API ([arXiv:1304.5990](https://arxiv.org/abs/1304.5990)).
+  The 7-strand braid group is CAT(0): Jeong,
+  [arXiv:2009.09350](https://arxiv.org/abs/2009.09350) (2020) — verified via arXiv
+  API; I could *not* verify journal publication, so treat as a preprint claim.
+  For $n\ge 8$ strands: **open**.
+
+**Partial/structural results around the conjecture.**
+
+- Dual (Garside/BKL) complexes give finite $K(\pi,1)$s for finite type
+  (Brady–Watt, "$K(\pi,1)$'s for Artin groups of finite type", *Geom. Dedicata*
+  **94** (2002), 225–250, [DOI 10.1023/A:1020902610809](https://doi.org/10.1023/A:1020902610809)
+  — verified on Crossref), but their curvature (the Brady–McCammond orthoscheme
+  conjecture, *Algebr. Geom. Topol.* **10** (2010), 2277–2314, verified via
+  Crossref reference data) is open in general; HKS 2016 resolved it for the
+  bounded graded modular complemented lattices behind $n\le 6$ strands.
+- The stronger conjecture that the Deligne complex with Moussong metric is always
+  CAT(0) (Charney–Davis) is open; recent step: Goldman–Herron, "The Deligne
+  complex for the $B_3$ Artin group", [arXiv:2503.15820](https://arxiv.org/abs/2503.15820)
+  (2025), verified via arXiv API — Moussong metric CAT(0) for type $B_3$, a step
+  toward 3-dimensional Artin groups.
+- Huang, "On spherical Deligne complexes of type $D_n$",
+  [arXiv:2405.11374](https://arxiv.org/abs/2405.11374) (2024), abstract verified —
+  proves center/quasi-center results for 6-cycles in type-$D_n$ spherical Deligne
+  complexes, aimed at $K(\pi,1)$, *not* a CAT(0)-ness theorem for type $D_n$.
+- Weaker nonpositive-curvature frameworks: Haettel's CUB/injective-metric link
+  condition ([arXiv:2211.07857](https://arxiv.org/abs/2211.07857), verified via
+  arXiv API) applies to Artin complexes of Euclidean-type Artin groups (CUB is
+  weaker than CAT(0)); Huang–Osajda proved systolicity for 2-dimensional Artin
+  groups (*Math. Ann.* **374** (2019), 1311–1352, verified via Crossref reference
+  data). Systolic/Helly/CUB actions give many CAT(0)-like corollaries but do not
+  answer the question.
+- Consistency check that the problem is still open in 2024–2026: Bregman–
+  Libgober–Zhu ([arXiv:2411.18067](https://arxiv.org/abs/2411.18067), verified via
+  arXiv API) state they are "motivated by the question of whether braid groups are
+  CAT(0)"; no publication or preprint claiming a full solution (finite-type or
+  general) was found in arXiv searches through July 2026.
+
+## Work done
+
+1. **Source identification.** Located Q 2.16 in Bestvina's updated (July 2004)
+   list; transcription verified verbatim, attribution to Ruth Charney confirmed.
+2. **Verified literature triage** (above; every item checked against Crossref or
+   the arXiv API; publication status of Jeong's 7-strand paper flagged as
+   unverified).
+3. **Reduction analysis (why finite type is the crux).** The natural candidate
+   space for any Artin group $A_\Gamma$ is its (modified) Deligne complex
+   $D_\Gamma$ — the geometric realization of the poset of cosets of spherical
+   standard parabolics — with the Moussong piecewise-Euclidean metric. $A_\Gamma$
+   acts on $D_\Gamma$ geometrically, and by Gromov's link condition $D_\Gamma$ is
+   CAT(0) iff every vertex link is CAT(1). Those links are (joins of) *spherical
+   Deligne complexes*, i.e. the corresponding complexes for the finite-type
+   parabolics $A_T$, $T\subseteq S$. Hence the Moussong-metric form of the
+   general question reduces by induction on rank to the statement:
+   *every spherical Deligne complex is CAT(1)* — a question purely about
+   finite-type Artin groups. This is exactly the bracketed case of Q 2.16, and it
+   is where all known obstructions live: the spherical Deligne complex of type
+   $A_{n-1}$ is the non-crossing partition (diagonal-link) complex $NC_n$, whose
+   CAT(1)-ness is known only for $n\le 7$ (HKS for $n\le 6$ via embeddability of
+   diagonal links into spherical buildings of type $A$; Jeong for $n=7$), and the
+   building-embedding method used up to $n=7$ provably does not extend naively —
+   new local-to-global phenomena (short loops without centers in the 1-skeleton)
+   appear. Type $B_n$, $D_n$, $F_4$, $E_{6,7,8}$, $H_4$ spherical complexes are
+   even less understood (only rank $\le 3$ cases and $B_3$ are settled, by Brady
+   2000 and Goldman–Herron 2025).
+4. **Obstruction check.** I looked for potential negative evidence (a finite-type
+   Artin group that is not CAT(0)): none exists in the literature; all structural
+   results (biautomaticity in low rank, Garside structure, injective/Helly
+   metrics, Farrell–Jones and Baum–Connes consequences in XXL cases) are
+   consistent with a positive answer. The community expectation (e.g. Haettel
+   2022) is that the answer is positive.
+
+## Result
+
+**Open.** No solution is claimed here; the contribution is a rigorous, verified
+literature triage plus a precise reduction: via the Deligne complex with the
+Moussong metric and Gromov's link condition, "all Artin groups are CAT(0)" reduces
+to "all spherical Deligne complexes are CAT(1)", which is exactly the finite-type
+case of the question. Current frontier:
+
+- braid group on $n$ strands: CAT(0) for $n\le 7$ (HKS 2016; Jeong 2020,
+  preprint), **open for $n\ge 8$**;
+- type $B_n$ ($n\ge 4$), $D_n$ ($n\ge 4$), $F_4$, $E_6,E_7,E_8$, $H_4$: **open**;
+  rank $\le 3$ and $B_3$ (Moussong metric) settled;
+- non-spherical: RAAGs, FC type, 2-dimensional, XXL ($m\ge 5$) are CAT(0);
+  large ($m\ge 3$) and extra-large ($m\ge 4$) type in rank $\ge 4$, and the
+  general case: **open**.
+
+## What remains
+
+- Decide CAT(1)-ness of the type-$A_{n}$ spherical Deligne (non-crossing
+  partition) complex for $n\ge 7$ (equivalently, braid groups on $\ge 8$
+  strands); the HKS building-embedding technique needs a new idea past $n=7$.
+- The same for types $B_n$, $D_n$ (Huang's 6-cycle/quasi-center analysis is a
+  step toward $K(\pi,1)$ but not yet a CAT(1) statement) and the exceptional
+  types $F_4, E_6, E_7, E_8, H_4$.
+- Even a positive answer to the bracketed (finite-type) question would not
+  immediately give the general case: the induction above only proves the
+  *Moussong metric on the Deligne complex* is CAT(0), which already fails for
+  some 2-dimensional Artin groups (Brady–Crisp); those were handled by different
+  3-dimensional complexes, and a uniform construction for all Artin groups is
+  missing.
+- Related weaker targets that are open and would be strong evidence: the
+  Brady–McCammond orthoscheme conjecture (CAT(0)-ness of dual Garside
+  complexes), and whether every Artin group admits a proper (not necessarily
+  cocompact) action on a CAT(0) space.
+- Verification gap to close: publication status of Jeong's 7-strand braid group
+  preprint (arXiv:2009.09350) could not be confirmed.