Cohomological and geometric invariants of simple complexes of groups

We investigate cohomological properties of fundamental groups of strictly developable simple complexes of groups X. We obtain a polyhedral complex equivariantly homotopy equivalent to X of the lowest possible dimension. As applications, we obtain a simple formula for proper cohomological dimension of CAT.0/ groups whose actions admit a strict fundamental domain; for any building of type (W,S) that admits a chamber transitive action by a discrete group, we give a realisation of the building of the lowest possible dimension equal to the virtual cohomological dimension of W; under general assumptions, we confirm a folklore conjecture on the equality of Bredon geometric and cohomological dimensions in dimension one; finally, we 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.