Abstract. We prove that if Γ is a lattice in a classical simple Lie group G, then the symmetric space of G is Γ-equivariantly homotopy equivalent to a proper cocompact Γ-CW complex of dimension the virtual cohomological dimension of Γ.

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

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.

Let G be a group that admits a cocompact classifying space for proper actions X. We derive a formula for the Bredon cohomological dimension for proper actions of G in terms of the relative cohomology with compact support of certain pairs of subcomplexes of X. We use this formula to compute the Bredon cohomological dimension for proper actions of fundamental groups of non-positively curved simple complexes of finite groups. As an application we show that if a virtually torsion-free group acts properly and chamber transitively on a building, its virtual cohomological dimension coincides with its Bredon cohomological dimension. This covers the case of Coxeter groups and graph products of finite groups.

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