The lower algebraic \(K\)-theory of virtually infinite cyclic groups
type: Item
instance of: scholarly article
title: The lower algebraic \(K\)-theory of virtually infinite cyclic groups
zbMATH Open document ID: 0829.19002
DOI: 10.1007/BF00965457
published in: \(K\)-Theory
publication date: 1996-01-17T00:00:00Z
review text: In an earlier paper [Isomorphism conjectures in algebraic \(K\)-theory, J. Am. Math. Soc. 6, No. 2, 249-297 (1993; Zbl 0798.57018)], the authors conjectured that \(K\)-theory for an arbitrary group \(\Gamma\) is ``computable'' from \(K\)-theory of the family of virtually cyclic subgroups of \(\Gamma\). The present paper deals with the ``atoms'', i.e. the virtually cyclic groups. For such a group, it is shown that \(K_n (\mathbb{Z} \Gamma)\) vanishes for \(n \leq -2\) and that \(K_{-1} (\mathbb{Z} \Gamma)\) is generated by the subgroups \(\text{Im} (K_{-1} (\mathbb{Z} F) \to K_{-1} (\mathbb{Z} \Gamma))\) with \(F\) varying over all finite subgroups of \(\Gamma\). The authors first treat the semi-direct product case \(\Gamma = G \rtimes T\) where \(G\) is finite and \(T\) infinite cyclic. Here the methods are algebraic, relying on techniques developed by Bass, Carter and Connolly and da Silva. The extension to the general case uses controlled topology and dynamics arguments developed by the authors in a series of earlier papers.
reviewed by: Hans Jørgen Munkholm
Mathematics Subject Classification ID: 19D35
zbMATH DE Number: 784412
zbMATH Keywords: \(\mathbb{Z}\)-order
zbMATH Keywords: \(A\)-lattice
zbMATH Keywords: pseudoisotopy
zbMATH Keywords: hyperbolic space
zbMATH Keywords: virtually cyclic groups
zbMATH Keywords: controlled topology
author: F. Thomas Farrell
author: Lowell Jones
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00965457
OpenAlex ID: W2028735900
description: scientific article; zbMATH DE number 784412