$K$– and $L$–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by $\mathbb{Z}/4$

Abstract

We compute the group homology, the topological $K$–theory of the reduced $C^{\ast }$–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 $\mathbb{Z}∕4$. These computations will follow from the more general treatment of a certain class of groups $G$ which occur as extensions $1\to K\to G\to Q\to 1$ 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.