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 $H_d(L;\mathbb{Z}/2)\ne 0$, 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.

Minor corrections

References in corpus (1)

• The action dimension of Artin groups

Cited by in corpus (5)

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