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

Lueck, Wolfgang

doi:10.2140/gt.2005.9.1639

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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

DOI: 10.2140/gt.2005.9.1639

arXiv: math.KT/0412156

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 $1 \to K \to G \to Q \to 1$ 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.

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

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Publication

Received: 8 December 2004

Accepted: 19 August 2005

Published: 28 August 2005

Proposed: Gunnar Carlsson

Seconded: Ralph Cohen, Bill Dwyer

Authors

Wolfgang Lueck
Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany
www.math.uni-muenster.de/u/lueck/

Volume 9, issue 3 (2005)