Bestvina complex for group actions with a strict fundamental domain

DOI10.4171/GGD/581arXiv1712.07606MaRDI QIDQ6295515

Nansen Petrosyan, Tomasz Prytuła

Publication date: 20 December 2017

Abstract: We consider a strictly developable simple complex of finite groups $G(mathcalQ)$. 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 $G(mathcalQ)$ 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.

Mathematics Subject Classification ID

Geometric group theory (20F65) Cohomology of groups (20J06) Reflection and Coxeter groups (group-theoretic aspects) (20F55) Homological methods in group theory (20J05) Groups acting on trees (20E08) Combinatorial aspects of simplicial complexes (05E45)

This page was built for publication: Bestvina complex for group actions with a strict fundamental domain

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6295515)