Is every word-hyperbolic group CAT(0)?
A group is CAT(0) (respectively CAT(-1)) if it acts geometrically, that is properly, cocompactly and by isometries, on a proper complete CAT(0) (respectively CAT(-1)) space. Word-hyperbolic groups are characterised by a linear isoperimetric inequality, a coarse form of negative curvature, and Gromov sought geometric realisations of them by negatively curved spaces. Free groups, closed hyperbolic manifold groups and many cubulated groups are CAT(0); Brady and Crisp found hyperbolic groups with no such action in dimension two. Does every hyperbolic group act geometrically on a proper CAT(0) space, or even a proper CAT(-1) space?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometric group theory; CAT(0) and CAT(-1) groups
- Posed by
- M. Gromov (realisation question); CAT(-1) form as Problem H11 in Baumslag-Myasnikov-Shpilrain, Open problems in combinatorial group theory (1999); both forms in E. Stark (2025)
- Year posed
- 1999
- Years open
- 27y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a finite connected two-dimensional aspherical simplicial complex with linear combinatorial disk filling, so is word-hyperbolic with a finite classifying space, such that has no geometric action on any proper complete CAT(0) space in any dimension. Hence it is neither CAT(0) nor CAT(-1), and no finite complex homotopy equivalent to carries a locally CAT(0) metric inducing its topology. The group comes from a random-presentation construction with a Fano-plane incidence core and a harmonic-energy fixed-point obstruction. Not shown: an explicit presentation, or which natural classes of hyperbolic groups (for example random or small-cancellation groups) are CAT(0).
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge HyperbolicObstruction), with scope limits listed in the release's Lean notes.
Verification
No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the CAT(0) and CAT(-1) realisation questions as cited. Lean-checked on the release's Comparator challenge HyperbolicObstruction (declaration OAI.HyperbolicObstruction.main, listed in lean/formalization.yaml). Its statement was read here: there is a finite simplicial complex , connected and aspherical, with a linear disk-filling bound, whose fundamental group satisfies four-point word hyperbolicity for some finite generating set and admits no geometric action on any nonempty proper complete CAT(0) space (CAT(0) by the CN inequality), and no finite complex homotopy equivalent to carries a compatible locally CAT(-1) geodesic metric. That covers the headline. As the release's own Lean notes say, it omits that is two-dimensional and that no such model is locally CAT(0). Not rebuilt here. The existence argument gives no effective size for .