Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

TL;DR

This paper investigates the asymptotic dimension of Artin and Coxeter groups, establishing new bounds related to the clique number of their defining graphs, and conjectures a general inequality for all such groups.

Contribution

It introduces a new upper bound for the asymptotic dimension of Artin and Coxeter groups based on the clique number, and proves the bound for specific cases, advancing understanding of their geometric properties.

Findings

Proves that if the conjecture holds for free of infinity Artin groups, it holds for all.

Establishes that the asymptotic dimension of Coxeter groups is bounded by the clique number.

Shows that Artin groups of large type with clique number 3 have asymptotic dimension exactly two.

Abstract

If $A_\Gamma$ ($W_\Gamma$) is the Artin (Coxeter) group with defining graph $\Gamma$ we denote by $Sim(\Gamma)$ the number of vertices of the largest clique in $\Gamma$. We show that $asdimA_\Gamma \leq Sim(\Gamma)$, if $Sim(\Gamma)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_\Gamma \leq Sim(\Gamma)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any Artin group of large type with $Sim(\Gamma)=3$ is exactly two.