#### Volume 9, issue 3 (2005)

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

### Wolfgang Lueck

Geometry & Topology 9 (2005) 1639–1676
 arXiv: math.KT/0412156
##### 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 $ℤ∕4$. 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.

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##### Keywords
$K$– and $L$–groups of group rings and group $C^*$–algebras, three-dimensional Heisenberg group
##### Mathematical Subject Classification 2000
Primary: 19K99
Secondary: 19A31, 19B28, 19D50, 19G24, 55N99