1 · The essentials
The manuscript answers a question about whether two ways of describing a mathematical system of symmetries always fit together. One describes the system as hyperbolic; the other asks whether it can act on a space with a particular kind of nonpositive curvature. The answer is no under the precise requirements in the manuscript. It constructs a system that satisfies the first description but cannot act on any space of the required kind, in any dimension. This system nevertheless has a finite topological description of the kind specified in the result. The manuscript also draws a consequence: an efficient way of filling loops in a finite space does not always allow that space to be replaced by a finite nonpositively curved model with the same underlying topological information. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a finite instruction booklet describing a system of moves, and a demand that those moves be realized in a room obeying strict geometric rules. The construction supplies a booklet for which no suitable room works, even if the room is allowed arbitrarily many dimensions. The image is limited: the mathematical obstruction concerns a precisely specified kind of action, not every possible realization.
Why it matters
The result gives a negative answer to whether every word-hyperbolic group is a CAT(0) group in the sense specified here. Its topological consequence also separates a linear disk-filling condition from the existence of a finite locally CAT(0) homotopy model. Thus, the filling condition alone does not guarantee either that model or a finite locally CAT(−1) model.
2 · Going further
The precise question is whether every word-hyperbolic group admits a geometric action on a proper complete CAT(0) space. Here a geometric action means a proper, cocompact action by isometries. The main manuscript constructs a hyperbolic group with a finite classifying space for which no such action exists. The repository summary explicitly states that the exclusion applies in any dimension.
The argument is presented in the abstract as a construction of a counterexample, rather than as a general classification. Its key reported combination is hyperbolicity and a finite classifying space together with an obstruction to every geometric action on the specified CAT(0) spaces. The supplied abstract does not describe the construction or the mechanism of that obstruction, so a more detailed account of the argument is not available from this source.
As a consequence, the manuscript shows that a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model. It consequently need not have a finite locally CAT(−1) model either. The metric models in this statement carry geodesic length metrics inducing the given complex topology. Only the main manuscript is supplied; no companion manuscripts or relations to them are described.
Useful notions
- Groups and actions by symmetries
- Word-hyperbolic groups
- CAT(0) metric geometry
- Classifying spaces and homotopy
- Combinatorial filling of loops
What was known before / what is new
Before
The repository summary identifies the question being answered: whether every word-hyperbolic group is a CAT(0) group. The supplied material gives no earlier partial results or historical attribution. It also identifies linear combinatorial disk filling as the condition relevant to the topological consequence.
New
The main result constructs a hyperbolic group with a finite classifying space but no geometric action on any proper complete CAT(0) space, in any dimension. The manuscript also shows that the stated linear disk-filling condition does not ensure a finite locally CAT(0) homotopy model, and hence does not ensure a finite locally CAT(−1) model. The main result has a Lean formalization.
Limits and assumptions
The nonexistence statement concerns proper, cocompact, isometric actions on proper complete CAT(0) spaces; it is not a claim that all actions on all CAT(0) spaces are impossible. The topological consequence concerns finite aspherical simplicial complexes satisfying a linear combinatorial disk-filling inequality, with model metrics required to be geodesic length metrics inducing the complex topology. The main result is Lean-formalized, but the supplied material does not specify whether the formalization also covers every stated consequence. No remaining open problems are identified in the supplied abstract.
Mini glossary
- Geometric action —
- A proper, cocompact action by isometries—the precise kind of action excluded in the main result.
- CAT(0) space —
- A geodesic metric space satisfying a nonpositive-curvature comparison condition for triangles.
- Finite classifying space —
- A finite topological model for a group whose fundamental group is that group and whose universal cover is contractible.
- Linear combinatorial disk-filling inequality —
- A bound allowing combinatorial loops to be filled with disks whose number of faces is at most a fixed constant times the loop length.
- Homotopy model —
- A space homotopy equivalent to the original space, retaining its homotopy information while potentially carrying different geometry.
Keywords : Hyperbolic groups · CAT(0) actions · Disk-filling inequalities
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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 space, in any dimension.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A hyperbolic group with no geometric CAT(0) action
25/09/2026We 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.