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Math-1800 · mathematics · Group theory

Questions in Geometric Group Theory — Q 2.16

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Statement

Definitions and mathematical statement

For a finite set $S$ with symmetric labels $m_{st}\in\{2,3,\ldots\}\cup\{\infty\}$, the Artin group $A$ equates the two alternating words of length $m_{st}$ in $s,t$ whenever that label is finite. No relation is imposed for an infinite label. The associated Coxeter group $W$ is the quotient obtained by adding $s^2=1$ for every $s\in S$. The Artin group has finite, or spherical, type when $W$ is finite; this does not mean that $A$ itself is finite.

A group is a CAT$(0)$ group if it admits an isometric action on a proper geodesic CAT$(0)$ space $X$ which is proper and cocompact. Properness of $X$ means compactness of closed bounded balls. The CAT$(0)$ condition is Euclidean comparison for geodesic triangles. Properness of the action means that only finitely many translates of any compact set meet that set.

Is every finite-type Artin group a CAT$(0)$ group? More broadly, ask the same for all finite defining matrices, as suggested by the source's bracketed finite-type restriction. The space and action may depend on the group; no uniform dimension, fixed cell shape, or prescribed complex is requested. A complete answer must produce a proper cocompact action, rather than merely an action with unbounded orbits or a locally nonpositively curved space whose fundamental group is a different group.

Short English statement

Do finite-type Artin groups, and more generally arbitrary Artin groups, admit geometric CAT(0) actions?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1800-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1800

The main 116-generator example refutes the broader finite-rank/proper-target version. It is not a spherical-type counterexample, so the separately requested finite-type version remains unanswered. This entry records partial support; complete coverage of the question is not established by this catalogue.

Questions in Geometric Group Theory — Q 2.16

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. A specific Artin group on 116 generators is claimed to have no proper cocompact isometric action on a proper CAT$0$ space.

Mathematical statement of the claimed result

Result 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.

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.

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. 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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