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Math-1796 · mathematics · Group theory

Questions in Geometric Group Theory — Q 2.9

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Statement

Definitions and mathematical statement

Let $S$ be a finite set and let $m_{st}=m_{ts}\in\{2,3,\ldots\}\cup\{\infty\}$ for distinct $s,t\in S$. The associated Artin group is \[ A=\langle S\mid \underbrace{stst\cdots}_{m_{st}\text{ letters}} =\underbrace{tsts\cdots}_{m_{st}\text{ letters}} \text{ whenever }m_{st}<\infty\rangle. \] An infinite label imposes no relation on that pair. No relations $s^2=1$ are included; imposing those would instead produce the associated Coxeter group.

A finite $K(A,1)$ is a connected CW complex $X$ with finitely many cells, fundamental group isomorphic to $A$, and contractible universal cover. Equivalently, $X$ is aspherical: its homotopy groups in every degree at least two vanish. Does every Artin group defined by finite data as above admit a finite $K(A,1)$?

Finiteness means a finite total number of cells, not merely finitely many cells in each dimension or a finite-dimensional complex. The desired complex need not be the standard presentation complex, whose higher homotopy may be nontrivial, nor any particular complex constructed from the defining diagram. Thus proving asphericity of a canonical candidate would suffice, but the question permits other constructions. The phrase Artin group here includes arbitrary finite defining matrices, not just those whose associated Coxeter group is finite. A solution for one diagram family therefore does not address the entire quantified class.

Short English statement

Does every Artin group with finitely many standard generators admit a finite aspherical classifying space?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1796-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1796

The canonical Salvetti complex is finite because a finite generator set has finitely many spherical subsets; the main theorem proves it is a classifying space, a specified model stronger than unspecified existence. The claimed conclusion supplies the requested case and extends it in the directions just specified.

Questions in Geometric Group Theory — Q 2.9

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Every finite-rank Artin Salvetti complex is claimed aspherical, allowing arbitrary Coxeter labels.

Mathematical statement of the claimed result

Result 1.
Choose an arbitrary Coxeter matrix $M$ on a finite set $S$, allowing infinite labels. Let $W$ be its Coxeter group and $X(W,S)$ the Artin Salvetti complex, with a $|T|$-cell for each spherical subset $T\subseteq S$. The source claims \[ \widetilde{X(W,S)}\text{ contractible},\qquad \pi_n(X(W,S))=0\quad\text{for all }n\ge2. \] No connectedness condition is imposed on the Coxeter diagram.

Outline of the argument

A framed category records spherical parabolic cosets and the Artin action. Weighted harmonic heights and two-point estimates organize convex layers; cohomological tests on those layers remove the higher homotopy obstruction and establish the classifying-space property.

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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