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

TL;DR

This paper introduces a method to determine the action dimension of Artin groups using complexes of abelian subgroups, extending previous results to more general cases and establishing bounds based on the nerve's topology.

Contribution

It generalizes the calculation of the action dimension for Artin groups by using complexes of abelian subgroups and relates it to the topology of the nerve complex.

Findings

actdim A or many cases, the minimal manifold dimension is 2d+2.

If the top homology of the nerve with  coefficients is non-zero, then actdim A or sure is at least 2d+2.

When the K(,1) conjecture holds, the lower bound is achieved, giving an exact value.

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.