Free subgroups and decompositions of one-relator products of cyclics. I: The Tits alternative
type: Item
instance of: scholarly article
title: Free subgroups and decompositions of one-relator products of cyclics. I: The Tits alternative
zbMATH Open document ID: 0639.20016
DOI: 10.1007/BF01194564
author: Benjamin Fine
author: Frank Levin
author: Gerhard Rosenberger
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \(G=<a_ 1,...,a_ n\); \(a_ i^{e_ i}=1\) (1\(\leq i\leq n)\), R \(m(a_ 1,...,a_ n)=1>\), where m,n\(\geq 2\), \(e_ i=0\) or \(e_ i\geq 2\) (1\(\leq i\leq n)\), and \(R(a_ 1,...,a_ n)\) is a cyclically reduced word in the free product in \(a_ 1,...,a_ n\) which involves all \(a_ 1,...,a_ n\). The authors prove that the Tits alternative holds for G (i.e. G has either a free subgroup of rank 2 or a solvable subgroup of finite index) in the following cases: (i) \(n\geq 3\), (ii) \(n=2\) and \((e_ 1=0\) or \(m\geq 2)\). The proofs involve representing G in \(PSL_ 2({\mathbb{C}})\), based on an earlier method of Baumslag, Morgan and Shalen.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 4042144
zbMATH Keywords: one relator product of cyclic groups
zbMATH Keywords: free product
zbMATH Keywords: Tits alternative
zbMATH Keywords: free subgroup
zbMATH Keywords: solvable subgroup of finite index
reviewed by: A. W. Mason
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4042144