By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality ℵn admits a finite-dimensional classifying space with virtually cyclic stabilizers of dimension n + h + 2. We also provide a criterion for groups that fit into an extension with torsion-free quotient to admit a finite-dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for proper actions. This answers a question posed by W. Lück. Contents

We compute the cohomology of crystallographic groups Γ = Z n Z/p with holonomy of prime order by establishing the collapse at E 2 of the spectral sequence associated to their defining extension. As an application we compute the group of gerbes associated to many six-dimensional toroidal orbifolds arising in string theory.

Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g ≥ 2 admits a (9g − 8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean buildings. In particular, we show that all finitely generated linear groups of positive characteristic have a finite dimensional classifying space for proper actions and a finite dimensional classifying space for the family of virtually cyclic subgroups. We also show that every generalized Baumslag-Solitar group has a 3-dimensional model for the classifying space with virtually cyclic stabilizers.

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for H F -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.