Is every word-hyperbolic group CAT(0)?

Theorem 1.1: there is a finite connected two-dimensional aspherical simplicial complex $K$ with linear combinatorial disk filling, so $\pi_1K$ is word-hyperbolic with a finite classifying space, such that $\pi_1K$ 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 $K$ 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).

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Is every word-hyperbolic group CAT(0)? · Hyperbolic groups and CAT(0) actions

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Is every word-hyperbolic group CAT(0)?

specialist registry · passed

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 $K$ , 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 $K$ carries a compatible locally CAT(-1) geodesic metric. That covers the headline. As the release's own Lean notes say, it omits that $K$ is two-dimensional and that no such model is locally CAT(0). Not rebuilt here. The existence argument gives no effective size for $K$ .