On the vanishing of the lower K-theory of the holomorph of a free group on two generators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.19003
author: Vassilis Metaftsis
author: Stratos Prassidis
published in: Forum Mathematicum
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.4139
Mathematics Subject Classification ID: 19D35
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 6072033
zbMATH Keywords: lower K-theory
zbMATH Keywords: holomorph-free groups
zbMATH Keywords: Farrell-Jones isomorphism conjecture
zbMATH Keywords: virtually cyclic groups
zbMATH Keywords: Whitehead group
zbMATH Keywords: fibered isomorphism conjecture
zbMATH Keywords: pseudoisotopy functor
MaRDI profile type: Publication
OpenAlex ID: W2963828081
arXiv ID: 1003.4139
cites work: A formula for the Whitehead group of a three-dimensional crystallographic group
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cites work: Isomorphism Conjectures in Algebraic K-Theory
cites work: Q4515188
cites work: Cerf Theory for Graphs
cites work: Contractibility of fixed point sets of auter space
DOI: 10.1515/FORM.2011.088
title: On the vanishing of the lower K-theory of the holomorph of a free group on two generators
review text: Let \(F_2\) be the free product on two generators. The holomorph group, \(\mathrm{Hol}(F_2)\), of \(F_2\) is the universal split extension of \(F_2\) by \(\mathrm{Aut}(F_2)\), hence it fits in an extension \(1\to F_2\to \mathrm{Hol}(F_2) \to \mathrm{Aut}(F_2)\to 1\). The authors prove in this paper that the fibered isomorphism conjecture for the pseudoisotopy functor is valid for the group \(\mathrm{Hol}(F_2)\). They also classify up to isomorphism the virtually cyclic subgroups that appear as subgroups of \(\mathrm{Hol}(F_2)\). The list of such groups is very limited, for example, the only infinite virtually cyclic subgroup of \(\mathrm{Hol}(F_2)\) is isomorphic to \(\mathbb{Z}/2\times \mathbb{Z}\). The main theorem is the following:NEWLINENEWLINE Main Theorem. The group \(\mathrm{Hol}(F_2)\) satisfies the Farrell-Jones fibered isomorphism conjecture for the pseudoisotopy functor. Furthermore, for any subgroup \(\Gamma<\mathrm{Hol}(F_2)\) it follows that \( Wh(\Gamma)=\widetilde{K}_0(\mathbb{Z}[\Gamma])=K_i(\mathbb{Z}[\Gamma])=0\) for \(i\leq -1\).
reviewed by: Daniel Juan Pineda
description: scientific article; zbMATH DE number 6072033