Determining the action dimension of an Artin group by using its complex of abelian subgroups

Abstract

Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and with a simplicial complex $L$ as its nerve. We define the notion of a "standard abelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized by the poset of simplices in a certain subdivision $L_\oslash$ of $L$. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right-angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for $BA$. (This is the "action dimension" of $A$ denoted actdim $A$.) If $H_d(L; \mathbb Z/2)\neq 0$, where $d=\dim L$, then actdim $A \ge 2d+2$. Moreover, when the $K(\pi,1)$-Conjecture holds for $A$, the inequality is an equality.