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A hyperbolic group with no geometric CAT(0) action
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Abstract
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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Martino Lupini, Robust fixed points and finite classifying spaces: CAT(0) actions, Banach targets and group fillings (Zenodo, doi:10.5281/zenodo.23424960, v2, 2026-10-16; v1 2026-10-14 under a different title). Starting from "OpenAI's group construction, diagram argument and analytic obstruction", it claims infinite torsion-free hyperbolic groups with finite two-dimensional classifying spaces whose isometric actions on every nonempty complete CAT(0) space have a fixed point, which it says answers an existence question recorded by Eskenazis, Mendel and Naor (2019).
Formalization
- OpenAI math release, Lean scope notes for family 257 (lean/docs/257.md, 2026-10-06). The formalized statement gives a finite connected aspherical complex with linear disk filling whose word-hyperbolic fundamental group has no geometric action on a nonempty proper complete CAT(0) space, and excludes finite locally CAT(-1) homotopy models; two-dimensionality and the exclusion of locally CAT(0) models are not formalized.
- OpenAI, HyperbolicObstruction.lean (openai/math, lean/ComparatorChallenges, 2026). The comparator statement file linked from the scope notes; it states
MainStatement, andmainin this file is closed bysorry.
Related prior literature mentioned by others
- Noel Brady and John Crisp, CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups (Commentarii Mathematici Helvetici 82, 61–85, 2007). AIProofWatch's discussion of this manuscript identifies these examples, which exclude proper actions on proper two-dimensional CAT(0) spaces, as predecessors to the manuscript's claimed obstruction in every dimension.
Possibly relevant literature (found by the agent, not checked)
- Mladen Bestvina (collector), Questions in Geometric Group Theory (problem list, revised 2000, updated July 2004). Q 1.5 (attributed to Davis) asks whether the Rips complex of a word-hyperbolic group carries an equivariant negatively curved metric for large d, which relates to the CAT(-1) consequence in Theorem 1.1.
- Koji Fujiwara and Tetsu Toyoda, Random groups have fixed points on CAT(0) cube complexes (arXiv:1012.4147, 2010). A fixed-point result for random groups acting on CAT(0) cube complexes, a target class excluded by Theorem 1.1.
- Assaf Naor and Lior Silberman, Poincaré inequalities, embeddings, and wild groups (arXiv:1005.4084, Compositio Math. 147 (2011) 1546–1572). Gives fixed-point results for Gromov's expander-based random groups acting on metric spaces satisfying nonlinear Poincaré inequalities, related to the harmonic-energy fixed-point framework used in the paper's obstruction.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 257
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/257.md. The Lean statements have not been checked by people against the paper's claims.
Family 257: 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 $\mathop{\mathrm{CAT}}\nolimits (0)$ space, in any dimension.
This is the only paper in this family.