Geometric dimension of groups for the family of virtually cyclic subgroups

By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality ℵ n 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.