Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete \mathop{\mathrm{CAT}}\nolimits (0) space, in any dimension. (Lean)
Manuscript
A hyperbolic group with no geometric CAT(0) action
September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026}
We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.
Result 257 · Group theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md