Geometric CAT(0) actions for word-hyperbolic groups
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Does every word-hyperbolic group admit a proper cocompact isometric action on a nonempty proper complete CAT(0) space?Short English statement
Does every word-hyperbolic group admit a proper cocompact isometric action on a nonempty proper complete CAT(0) space?Sources
- S1 Original question source or mathematical context. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2713
The main theorem gives a word-hyperbolic fundamental group with no proper cocompact action on any proper complete CAT(0) space. This directly violates the target; the extra two-dimensional witness is unnecessary. lean/docs/257.md documents exactly the action obstruction even though it excludes the paper stronger dimension/local-CAT(0) details. Within these assumptions, this is a claimed negative answer to the stated question.
Geometric CAT$0$ actions for word-hyperbolic groups
OpenAI's claim
OpenAI presents a counterexample in the selected manuscript. A hyperbolic group with a finite classifying complex is claimed to admit no proper and cocompact isometric action with proper complete CAT$0$ target.Mathematical statement of the claimed result
Result 1.
The source constructs a finite, connected, aspherical simplicial complex $K$ of dimension two. In its universal cover, a length-$\ell$ combinatorial loop admits a triangular disk diagram with at most $C\ell$ faces, for one finite constant $C$. Nevertheless, \[ \pi_1K\text{ has no proper cocompact isometric action on a proper complete CAT}(0)\text{ space}. \] The linear filling estimate makes $\pi_1K$ word-hyperbolic. Within the homotopy class of $K$, no simplicial complex with finitely many simplices admits a geodesic length metric compatible with its topology and locally CAT$0$; the claim also excludes such a locally CAT$-1$ metric.Outline of the argument
Nested incidence tables produce finite relations with a linear disk-filling bound. Conditional sampling creates a limiting configuration of controlled energy, and its geometric obstruction contradicts any proposed CAT$0$ action; the finite aspherical model preserves the topological finiteness claim.Formal verification
For family 257, the documented construction gives a finite aspherical hyperbolic group model without a geometric proper-complete CAT$0$ action. The stronger dimension-two and all locally CAT$0$ model claims are excluded. The listed Comparator files areHyperbolicObstruction.lean. Named declarations include OAI.HyperbolicObstruction.main. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, A hyperbolic group with no geometric CAT$0$ action, 2026-09-25; citation key
OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 257: Lean scope.
- Comparator HyperbolicObstruction.json.
- Comparator HyperbolicObstruction.lean.
- OpenAI catalogue entry.