A hyperbolic group without a geometric CAT(0) action

A hyperbolic group that cannot act geometrically on any CAT(0) space, in any dimension.

1 paper September 25, 2026 Explained by AI

Hyperbolic groups behave like negatively curved spaces at large scales. Gromov looked for ways to realize each one by a space curved like that at every scale. This paper proves no, with a Lean formalization listed: it builds a hyperbolic group with no proper, cocompact action on any CAT(0) space. CAT(0) spaces have triangles at least as thin as flat ones.

First step

1987

Mikhail Gromov, whose foundational theory linked linear filling with hyperbolicity

Significance

Major

Whether every hyperbolic group acts geometrically on a CAT(0) or CAT(−1) space was a long-standing open problem.

Trust

Formal proof listed

Field

Group theory

1 paper

The explanation

Draw a square loop of side n on a flat grid of unit squares. Its length is 4n. To fill it you need n² unit squares. For n = 10 that is length 40 and area 100. For n = 100 it is length 400 and area 10,000. The area grows like the square of the length. Linear filling means any loop of length ℓ bounds a disk with at most Cℓ pieces, for a fixed C. Linear filling characterizes word-hyperbolic groups among finitely presented groupsSource 1. The flat grid is not hyperbolic.

A CAT(0) space is a space whose geodesic triangles are at least as thin as flat triangles. CAT(−1) means at least as thin as triangles in the hyperbolic plane. A geometric action is proper, cocompact and by isometriesSource 2. Proper means a compact region meets only finitely many of its moved copies. Cocompact means finitely many moved copies of one compact piece cover the space. The question asked whether every hyperbolic group has a geometric action on a proper CAT(0) or CAT(−1) spaceSource 3.

Many groups do. Free groups act on trees, and hyperbolic manifold groups act on hyperbolic spaceSource 4. Random groups at density below 1/6 act on CAT(0) cube complexesSource 5. Brady and Crisp found hyperbolic groups with no such action on a two-dimensional CAT(0) spaceSource 6. Their obstruction depends on the target's dimension. The obstacle was to block every dimension at once.

The paper proves the full obstruction. It builds a finite connected aspherical complex K with linear fillingSource 7. Aspherical means its universal cover can shrink to a point. The group of loops in K is word-hyperbolicSource 8. It has no geometric action on a proper complete CAT(0) spaceSource 9. The relations come from a modified incidence graph of the projective plane over the two-element fieldSource 10. That plane, the Fano plane, has 7 points and 7 lines, each line holding 3 points. Its incidence graph has 7 + 7 = 14 vertices and 7 × 3 = 21 edges. The relations are random but sparse, which keeps hyperbolicity. An energy argument then shows any action on a CAT(0) space would have a fixed pointSource 11. A geometric action of an infinite group cannot fix a point. The proof is an existence argument and gives no effective size for KSource 12.

How sure is this? A computer proof checker called Lean has a formal proof of the complex and its linear filling. It also covers hyperbolicity and the missing CAT(0) actions. It also excludes locally CAT(−1) metrics on equivalent finite complexes. The two-dimensional claim and the locally CAT(0) claim are not included. No specialist review of the paper has been published.

What it doesn't show. This does not give an explicit size for the complex. Lean does not cover the two-dimensional claim or the locally CAT(0) metric claim.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds, and the main obstruction has a Lean formalization. Some stronger claims are not in the Lean note. No specialist review is published.

Words used

Word-hyperbolic group: A finitely generated group whose distance graph has thin triangles; equivalently, loops have linear filling area.
Linear isoperimetric inequality: Every loop of length ℓ bounds a disk made of at most a constant times ℓ pieces.
CAT(0) space: A space whose geodesic triangles are at least as thin as triangles in a flat plane.
Geometric action: An action by isometries that is proper and cocompact.
Aspherical: Having a universal cover that can be shrunk to a point.
Fano plane: The projective plane over the two-element field, with 7 points and 7 lines of 3 points each.
Fixed point: A point that every element of the group leaves in place.

The original paper

A hyperbolic group with no geometric CAT(0) action (PDF, opens in a new tab)
September 25, 2026 Formal proof listed

Lean notes

All quoted lines (12)

A linear isoperimetric inequality is a large-scale expression of negative curvature: for finitely presented groups, it characterizes word hyperbolicity.
A hyperbolic group with no geometric CAT(0) action Back to text
Throughout the paper, a geometric action is a proper, cocompact action by isometries, and metric models of finite complexes carry geodesic length metrics inducing the given complex topology.
A hyperbolic group with no geometric CAT(0) action Back to text
The geometric realization question asks whether every hyperbolic group admits a geometric action on a proper $\CAT(0)$ space, or on a proper $\CAT(-1)$ space.
A hyperbolic group with no geometric CAT(0) action Back to text
Finitely generated free groups act on trees and closed hyperbolic manifold groups act on hyperbolic space.
A hyperbolic group with no geometric CAT(0) action Back to text
For example, Ollivier–Wise prove that random groups in the density model at density less than $1/6$ act freely and cocompactly on $\CAT(0)$ cube complexes with probability tending to one as relator length grows, with the number of generators fixed.
A hyperbolic group with no geometric CAT(0) action Back to text
Brady–Crisp construct infinitely many torsion-free hyperbolic groups of geometric dimension two which admit no proper isometric action on a proper two-dimensional $\CAT(0)$ space.
A hyperbolic group with no geometric CAT(0) action Back to text
Every combinatorial edge loop of length $\ell$ in the universal cover of $K$ bounds a disk diagram with at most $C\ell$ triangular faces.
A hyperbolic group with no geometric CAT(0) action Back to text
In particular, $\pi_1K$ is word-hyperbolic, and no finite simplicial complex homotopy equivalent to $K$ admits a locally $\CAT(0)$ geodesic length metric inducing its given topology.
A hyperbolic group with no geometric CAT(0) action Back to text
The group $\pi_1K$ admits no geometric action on a proper complete $\CAT(0)$ space.
A hyperbolic group with no geometric CAT(0) action Back to text
We organize the relations using a fixed modification of the incidence graph of the projective plane over $\mathbb F_2$ .
A hyperbolic group with no geometric CAT(0) action Back to text
In this framework, expansion in a finite graph can force a fixed point when the target satisfies a suitable nonlinear spectral inequality.
A hyperbolic group with no geometric CAT(0) action Back to text
The proof is an existence argument and does not give an effective size for $K$ .
A hyperbolic group with no geometric CAT(0) action Back to text

Other sources

openai/math, Lean scope note for family 257 whose fundamental group is word-hyperbolic but admits no geometric action on a nonempty proper complete $\mathrm{CAT}(0)$ space.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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