2022•arXiv (Cornell University)Open access
Asymptotic dimension of Artin groups and a new upper bound for Coxeter\n groups
Panagiotis Tselekidis
Open full text 0 citations
Abstract
If $A_\\Gamma$ ($W_\\Gamma$) is the Artin (Coxeter) group with defining graph\n$\\Gamma$ we denote by $Sim(\\Gamma)$ the number of vertices of the largest\nclique in $\\Gamma$. We show that $asdimA_\\Gamma \\leq Sim(\\Gamma)$, if\n$Sim(\\Gamma)=2$. We conjecture that the inequality holds for every Artin group.\nWe prove that if for all free of infinity Artin (Coxeter) groups the conjecture\nholds, then it holds for all Artin (Coxeter) groups. As a corollary, we show\nthat $asdimW_\\Gamma \\leq Sim(\\Gamma)$ for all Coxeter groups, which is the best\nknown upper bound for the asymptotic dimension of Coxeter Groups. As a further\ncorollary, we show that the asymptotic dimension of any Artin group of large\ntype with $Sim(\\Gamma)=3$ is exactly two.\n
Open-access reader