Bestvina complex for group actions with a strict fundamental domain

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.