Action dimensions of some simple complexes of groups

TL;DR

This paper calculates the minimal dimension of contractible manifolds on which certain groups act properly, focusing on direct limits of simple complexes of groups including Artin groups, graph products, and hyperplane arrangement groups.

Contribution

It provides explicit computations of action dimensions for several important classes of groups derived from simple complexes of groups.

Findings

Action dimension computed for Artin groups

Action dimension determined for graph products of groups

Action dimension established for fundamental groups of hyperplane arrangement complements

Abstract

The action dimension of a discrete group $G$ is the minimum dimension of contractible manifold that admits a proper $G$-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.