Virtually free groups and integral representations
type: Item
instance of: scholarly article
title: Virtually free groups and integral representations
zbMATH Open document ID: 1401.20025
author: Igor Lima
author: Pavel Zalesskij
published in: Journal of Algebra
publication date: 2018-02-28T00:00:00Z
review text: A virtually free group \(G\) is an extension of a (non abelian) free group \(F\) by a finite group \(H\). The action of \(H\) on \(F\) induces in a natural way the action of \(H\) on the abelization \(F^{\mathrm{ab}}=F/F'\) of \(F\) that turns \(F^{\mathrm{ab}}\) into a \(\mathbb{Z}[H]\)-module. The objective of this paper is to clarify some issues such as the structure of such a module, if, and in which cases, it is decomposable and determining its indecomposable components. By results of \textit{A. Karrass} et al. [J. Aust. Math. Soc. 16, 458--466 (1973; Zbl 0299.20024)], \textit{D. E. Cohen} [``Groups with free subgroups of finite index'', Lect. Notes Math. 319, 26--44 (1973; \url{doi:10.1007/BFb0058927})] and Scott \(G =\pi_{1}(\mathcal{G},\Gamma)\) is the fundamental group of a graph of finite groups. Then the authors describe the structure of \(\mathbb{Z}[H]\)-module \(F^{\mathrm{ab}}\) in terms of the graph of the group \(\pi_{1}(\mathcal{G},\Gamma)\). The main result (Theorem 1.1) is too technical to be stated here; an interesting consequence is the following corollary: Theorem 1.2. Let \(H\) be a finite \(p\)-group. Then \(F^{\mathrm{ab}}\) is a permutation \(\mathbb{Z}[H]\)-module if and only if \(G\) is an \(HNN\)-extension of \(H\) (with finitely many stable letters). Some interesting examples are also provided which illustrate the achieved results.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 6845383
zbMATH Keywords: virtually free groups
zbMATH Keywords: integral representations
zbMATH Keywords: Bass-Serre theory of groups acting on trees
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.02.015
OpenAlex ID: W2593412823
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DOI: 10.1016/J.JALGEBRA.2017.02.015
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description: scientific article; zbMATH DE number 6845383