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Math-1987 · mathematics · Group theory

Questions in Geometric Group Theory — Q 1.5

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Statement

Definitions and mathematical statement

Fix a finite symmetric generating set $S$ of a group $G$ and give its Cayley graph the path metric with edges of length one. The group is word-hyperbolic when there is $\delta\geq0$ such that every side of every geodesic triangle lies within distance $\delta$ of the other two sides.

For $d>0$, the Rips complex $P_d(G)$ is the simplicial complex whose vertex set is $G$ and whose finite simplices are exactly the sets $A\subset G$ satisfying \[ \operatorname{diam}(A)=\max_{a,b\in A}d_S(a,b)\leq d. \] Left multiplication gives a simplicial action of $G$ on this complex. A metric on its realization is equivariant here if this action is by isometries. A geodesic metric is CAT($-1$) if distances between points of each geodesic triangle are no greater than the corresponding comparison distances in the hyperbolic plane of curvature $-1$.

Does every word-hyperbolic $G$ have a threshold $d_0$ such that, for all sufficiently large Rips parameters $d\geq d_0$, $P_d(G)$ admits a compatible complete geodesic CAT($-1$) metric invariant under $G$? Equivalently one may allow a strictly negative curvature bound and rescale the metric. The requested metric may differ from the usual metric obtained by giving all simplices the same Euclidean shape; the underlying Rips complex and its specified action are retained.

Short English statement

Can the sufficiently large Rips complexes of every hyperbolic group themselves carry invariant negatively curved metrics?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1987-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1987

For the main hyperbolic example any such Rips metric would give a geometric CAT(0) action: the finite generating set makes the Rips complex locally finite with finite quotient, and a compatible complete locally compact geodesic metric is proper. This contradicts the core no-geometric-CAT(0) conclusion. Within these assumptions, this is a claimed negative answer to the stated question.

Questions in Geometric Group Theory — Q 1.5

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 are HyperbolicObstruction.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

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