## Geometry & Topology

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

Wolfgang Lueck

#### 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.

#### Article information

Source
Geom. Topol., Volume 9, Number 3 (2005), 1639-1676.

Dates
Accepted: 19 August 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799645

Digital Object Identifier
doi:10.2140/gt.2005.9.1639

Mathematical Reviews number (MathSciNet)
MR2175154

#### Citation

Lueck, Wolfgang. $K$– and $L$–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by $\mathbb{Z}/4$. Geom. Topol. 9 (2005), no. 3, 1639--1676. doi:10.2140/gt.2005.9.1639. https://projecteuclid.org/euclid.gt/1513799645

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