The lower algebraic \(K\)-theory of \(\Gamma_{3}\)
type: Item
instance of: scholarly article
title: The lower algebraic \(K\)-theory of \(\Gamma_{3}\)
zbMATH Open document ID: 1068.19004
published in: \(K\)-Theory
publication date: 2005-04-07T00:00:00Z
review text: Let \(\Gamma_3=O^+(3,1)\cap GL(4,\mathbb Z)\). The main theorem computes the lower algebraic K-theory of \(\Gamma_3\) as follows: \(\text{Wh}(\Gamma_3)=0\), \(\tilde K_0(\mathbb Z \Gamma_3)\cong \mathbb Z/4 \oplus\mathbb Z/4\), \(K_{-1}(\mathbb Z \Gamma_3)\cong \mathbb Z \oplus\mathbb Z\), \(K_{n}(\mathbb Z \Gamma_3)=0\) for \(n<-1\). The computation is based on the fact that \(\Gamma_3\) is a group for which the isomorphism conjecture of \textit{F. T. Farrell} and \textit{L. E. Jones} [J. Am. Math. Soc. 6, 249--297 (1993; Zbl 0798.57018)] is known to hold. This reduces the computation to an assembly of results for virtually cyclic subgroups. In fact there will be a further reduction to finite subgroups using algebraic techniques of \textit{J. F. Davis} and \textit{W. Lück} [K-Theory 15, 201--252 (1998; Zbl 0921.19003)]. The author explains all this and then proceeds to the actual computation, which is quite involved. To learn about all the virtually cyclic subgroups of \(\Gamma_3\) she relies heavily on the fact that \(\Gamma_3\) is a hyperbolic reflection group. Many earlier \(K\)-theory computations for group rings are needed before one can start assembling. The reduction to an assembly over finite subgroups also needs some explicit checks, valid only in the low degrees needed in the theorem. This is a very instructive case study.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 19B28
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 19D35
zbMATH DE Number: 2153602
zbMATH Keywords: hyperbolic \(n\)-simplex reflection groups
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: assembly map
author: Ivonne Johanna Ortiz
MaRDI profile type: Publication
full work available at URL: s10977-004-1484-9
OpenAlex ID: W1989720500
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DOI: 10.1007/S10977-004-1484-9
description: scientific article; zbMATH DE number 2153602