research/AMR-010-0118.md

metadata

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):

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 and arXiv: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 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 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): 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, 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 and the Oxford GGT book bibliography): 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 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 $${\check{H}}^k(\mathrm{\partial }G)\cong H^{k+1}(G;\mathbb{Z}G)\cong H_c^{k+1}(X)=\underset{R}{\mathrm{}}{H}^{k+1}(X,X\setminus B_R),\backslash check\; H^\{k\}(\backslash partial\; G)\backslash ;\backslash cong\backslash ;H^\{k+1\}(G;\backslash mathbb\{Z\}G)\backslash ;\backslash cong\backslash ;H^\{k+1\}\_c(X)\backslash ;=\backslash ;\backslash varinjlim\_\{R\}\backslash ,H^\{k+1\}(X,\backslash ,X\backslash setminus\; B\_R),$$ 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.