We consider a strictly developable simple complex of finite groups . 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 is non-positively curved, this implies that the Bestvina complex is a cocompact classifying space for proper actions of of minimal dimension. As an application, we show that for groups that act properly and chamber transitively on a building of type , the dimension of the associated Bestvina complex is the virtual cohomological dimension of . 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.
Cite this article
Nansen Petrosyan, Tomasz Prytuła, Bestvina complex for group actions with a strict fundamental domain. Groups Geom. Dyn. 14 (2020), no. 4, pp. 1277–1307DOI 10.4171/GGD/581