Computations of \(K\)- and \(L\)-theory of cocompact planar groups
type: Item
instance of: scholarly article
title: Computations of \(K\)- and \(L\)-theory of cocompact planar groups
zbMATH Open document ID: 0979.19003
DOI: 10.1023/A:1026539221644
author: Wolfgang Lück
author: Roland Stamm
published in: \(K\)-Theory
publication date: 2002-02-21T00:00:00Z
review text: Let \(G\) be either a cocompact planar group or an extension of the form \(1\to\mathbb{Z}^n\to G\to\pi\to 1\) where \(\pi\) is finite and acts freely on \(\mathbb{Z}^n\) outside 0. This paper contains explicit computations of the algebraic \(K\)-groups \(K_p(\mathbb{Z} G)\) for \(p\leq 1\), of the algebraic \(L\)-groups \(L_p(\mathbb{Z} G)\) for \(p\in \mathbb{Z}\) (usually with 2 inverted), and of the topological \(K\)-groups \(K_p(C^*_r)\), \(p\in \mathbb{Z}\), of the reduced \(C^*\)-algebra of \(G\). The computations apply in particular to two-dimensional crystallographic groups and to cocompact Fuchsian groups. The calculations are based on the facts that, for the groups under consideration, the finite subgroups and their normalizers are known and restricted in nature, and that the infinite virtually cyclic subgroups of cocompact groups can be listed. It is therefore possible to verify that certain assembly maps are isomorphisms, effectively confirming the Farrell-Jones and Baum-Connes isomorphism conjectures in this context.
Mathematics Subject Classification ID: 19G24
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 19B28
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 19D35
Mathematics Subject Classification ID: 19A31
zbMATH DE Number: 1563259
zbMATH Keywords: confirming of isomorphism conjectures
zbMATH Keywords: planar groups
zbMATH Keywords: algebraic \(L\)-groups
zbMATH Keywords: topological \(K\)-groups
zbMATH Keywords: two-dimensional crystallographic groups
zbMATH Keywords: cocompact Fuchsian groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563259