Controlled algebraic \(K\)-theory of integral group ring of \(SL(3,\mathbb{Z})\)
type: Item
instance of: scholarly article
title: Controlled algebraic \(K\)-theory of integral group ring of \(SL(3,\mathbb{Z})\)
zbMATH Open document ID: 0857.19002
DOI: 10.1007/BF00533217
author: Yanyan Li
published in: \(K\)-Theory
publication date: 1997-03-03T00:00:00Z
review text: The isomorphism-conjecture of Farrell and Jones states that the assembly map \[ \underset\text{Or}(G,{\mathcal F}{)}{\text{hocolim}} \mathbb{K}\mathbb{K}(G/G) \] in an equivalence of \(\Omega\)-spectra where \(G\) is a group, \({\mathcal F}\) is an appropriate family of subgroups, \(\text{Or}(G,{\mathcal F})\) is the orbit category with respect to \({\mathcal F}\) and \(\mathbb{K}\) is a functor from the full orbit category \(\text{Or}(G)\) to the category of \(\Omega\)-spectra. Farrell and Jones have proven this conjecture for the two-fold loops of the \(\Omega\)-spectrum given by topological pseudo-isotopy for \({\mathcal F}\) the family of virtually cyclic subgroups, provided that \(G\) is a cocompact discrete subgroup of a virtually connected Lie group. The paper deals with the computation of \(\pi_n\) of the source of the assembly map in this case for \(i=-1,0,1\), where \(G\) is a finitely generated infinite subgroup of \(SL(3,\mathbb{Z})\). It turns out to be zero for \(i=0,1\) and to be \(\mathbb{Z}^k\), where \(k\) is the number of distinct conjugacy classes of the dihedral group of order 12 in \(G\). In the first step the calculation is reduced to the family of finite subgroups and in the second step a spectral sequence argument is used.
reviewed by: Wolfgang Lück
Mathematics Subject Classification ID: 19B28
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 19B99
zbMATH DE Number: 929695
zbMATH Keywords: controlled \(K\)-theory
zbMATH Keywords: Quinn's spectral sequence
zbMATH Keywords: topological pseudo-isotopy
zbMATH Keywords: virtually cyclic subgroups
zbMATH Keywords: infinite subgroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 929695