K– andL–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by ℤ∕4

Abstract

We compute the group homology, the topological K -theory of the reduced C * -algebra, the algebraic K -theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1 → K → G → Q → 1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.