Result 257, Group theory

A hyperbolic group without a geometric CAT(0) action

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 CAT(0)\mathop{\mathrm{CAT}}\nolimits (0) space, in any dimension.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Groups encode geometries through their symmetries, but geometry that looks negatively curved at large scales need not admit a globally nonpositively curved realization. The unreviewed manuscript reports a counterexample to this proposed bridge.

What changes?

A word-hyperbolic group has uniformly thin triangles in its large-scale geometry. The manuscript constructs one with a finite classifying space, a finite complex encoding the group whose universal cover is contractible. Yet it has no geometric action on any proper complete CAT(0) space, in any dimension. CAT(0) means geodesic triangles are no fatter than Euclidean comparison triangles; "proper" means closed bounded sets are compact. A geometric action preserves distances, is proper, and has compact quotient.

What does that help mathematicians do?

The stated consequence concerns finite aspherical simplicial complexes, whose universal covers are contractible. Even if every combinatorial loop fills with a number of triangles bounded linearly by its length, such a complex need not be homotopy equivalent to a finite locally CAT(0) complex, nor therefore to a finite locally CAT(-1) complex. These models must have geodesic length metrics inducing their topology. Thus efficient disk filling alone cannot guarantee either local curvature model.

Are there practical applications?

Its immediate value is foundational: it separates large-scale negative curvature of groups from the existence of these geometric symmetry models. Researchers seeking CAT(0) models need assumptions beyond hyperbolicity and a finite classifying space. Those properties alone cannot justify transferring methods that require a geometric CAT(0) action.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A hyperbolic group with no geometric CAT(0) action

September 25, 2026 43 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026,
  author = {{OpenAI}},
  title = {{A hyperbolic group with no geometric CAT(0) action}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf}{OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/257.md.

A hyperbolic group without a geometric CAT(0) action

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result constructs a finite connected aspherical complex with a linear disk-filling bound whose fundamental group is word-hyperbolic but admits no geometric action on a nonempty proper complete CAT(0)\mathrm{CAT}(0) space. It also excludes locally CAT(−1)\mathrm{CAT}(-1) geodesic metrics on every finite complex homotopy equivalent to the witness. The stronger claims that the witness is two-dimensional and that those models admit no locally CAT(0)\mathrm{CAT}(0) metric are not included.

Comparator links

Result Comparator statement
Hyperbolic group with no geometric CAT(0) action HyperbolicObstruction.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.