research/AMR-010-0123.md

---
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.