Bestvina complex for group actions with a strict fundamental domain

TL;DR

This paper extends Bestvina's construction to a broader class of group actions, producing minimal-dimensional classifying spaces and providing new insights into cohomological properties of these groups.

Contribution

It generalizes Bestvina's complex to strictly developable complexes of groups, establishing its homotopy equivalence to the standard development and its role as a minimal classifying space.

Findings

Bestvina complex is homotopy equivalent to the standard development.

In non-positively curved cases, it provides a cocompact classifying space of minimal dimension.

The dimension of the Bestvina complex equals the virtual cohomological dimension for certain group actions.

Abstract

We consider a strictly developable simple complex of finite groups $G(\mathcal Q)$. We show that Bestvina's construction for Coxeter groups applies in this more general setting to produce a complex that is equivariantly homotopy equivalent to the standard development. When $G(\mathcal Q)$ is non-positively curved, this implies that the Bestvina complex is a cocompact classifying space for proper actions of $G$ of minimal dimension. As an application, we show that for groups that act properly and chamber transitively on a building of type $(W, S)$, the dimension of the associated Bestvina complex is the virtual cohomological dimension of $W$. We give further examples and applications in the context of Coxeter groups, graph products of finite groups, locally $6$-large complexes of groups and groups of rational cohomological dimension at most one. Our calculations indicate that, because of its minimal cell structure, the Bestvina complex is well-suited for cohomological computations.