Cohomological and geometric invariants of simple complexes of groups

Abstract

We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and give a realisation of the building of the lowest possible dimension. We introduce the notion of a reflection-like action, and use it to give a new family of counterexamples to the strong form of Brown's conjecture on the equality of virtual cohomological dimension and Bredon cohomological dimension for proper actions. We show that the fundamental group $G$ of a simple complex of groups acts on a tree with stabilisers generating a family of subgroups $\mathcal{F}$ if and only if its Bredon cohomological dimension with respect to $\mathcal{F}$ is at most one. This confirms a folklore conjecture under the assumption that a model for the classifying space $E_{\mathcal{F}}G$ of $G$ for the family $\mathcal{F}$ has a strict fundamental domain. In order to handle complexes of groups arising from arbitrary group actions, we define a number of combinatorial invariants such as the block poset, which may be of independent interest. We also derive a general formula for Bredon cohomological dimension for a group $G$ admitting a cocompact model for $E_{\mathcal{F}}G$. As a consequence of both, we obtain a simple formula for proper cohomological dimension of $\mathrm{CAT}(0)$ groups whose actions admit a strict fundamental domain.