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Math-1793 · mathematics · Group theory

Questions in Geometric Group Theory — Q 2.2

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Statement

Definitions and mathematical statement

A classifying space $K(G,1)$ is a connected CW complex with fundamental group $G$ and contractible universal cover. Say that $G$ has finite geometric dimension if it admits such a complex whose cells have dimensions bounded by some integer; the complex need not have finitely many cells or a finite skeleton.

A geodesic metric space is CAT$(0)$ if distances between points on sides of a geodesic triangle are at most the corresponding distances in its Euclidean comparison triangle. Completeness means that every Cauchy sequence converges. An isometric action of the discrete group $G$ is proper here if, for every compact subset $K$ of the space, the set \[ \{g\in G:gK\cap K\ne\varnothing\} \] is finite; this is the proper-discontinuity convention in the question.

Does every group with a finite-dimensional $K(G,1)$ admit such an action on some complete CAT$(0)$ space? The target may have infinite dimension and need not be locally compact or proper as a metric space. The action is not required to be cocompact, and the target need not arise by placing a metric on the supplied classifying space. These freedoms are part of the problem. In particular, an obstruction only to finite-dimensional, cocompact, or semisimple CAT$(0)$ actions would not settle the stated question. Neither finite generation nor finite presentation is separately imposed beyond what the hypothesis actually guarantees.

Short English statement

Does a finite-dimensional classifying space guarantee a proper isometric action on some complete CAT(0) space?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1793-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1793

The recorded question allows a proper isometric action without a cocompactness assumption. The selected Artin and hyperbolic examples rule out proper cocompact actions, with their indicated finite-dimensional classifying spaces. They do not rule out the noncocompact actions permitted here. These are related obstructions to a stronger action requirement; the existence question on this page remains unanswered by the selected theorems.

Questions in Geometric Group Theory — Q 2.2

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A specific Artin group on 116 generators is claimed to have no proper cocompact isometric action on a proper CAT$0$ space; A hyperbolic group with a finite classifying complex is claimed to admit no proper and cocompact isometric action with proper complete CAT$0$ target. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Manuscript 1.
For an Artin matrix $M$ on 116 generators with off-diagonal entries only in $\{2,3,\infty\}$, the constructed Artin group $A_M$ is claimed to have no action that is simultaneously isometric, proper, and cocompact on any nonempty proper CAT$0$ metric space. In quantified form, \[ \nexists X,\ A_M\curvearrowright X:\quad X\text{ proper CAT}(0),\quad\text{action proper and cocompact}. \] The metric space is required to be proper, and the exclusion is independent of dimension.
Manuscript 2.
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

Stable translation lengths and centralizer characters impose quantitative braid constraints. Six asymptotic sectors and labeled triangular diagrams yield a local winding count incompatible with those constraints; a finite Artin matrix realizes the contradictory configuration.

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 254, the scope includes finite-rank Salvetti contractibility, the 116-generator Artin CAT$0$ counterexample, and arbitrary parabolic intersections with closure consequences. Each has its own Comparator statement. The listed Comparator files are ArtinCAT0.lean, ArtinParabolicIntersections.lean, HarmonicArtin.lean. Named declarations include OAI.ArtinCAT0.main, OAI.HarmonicArtin.salvetti_cover_contractible. 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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