The universal torsion-free image of a group
type: Item
instance of: scholarly article
title: The universal torsion-free image of a group
zbMATH Open document ID: 0880.20023
DOI: 10.1007/BF02937335
author: Sergei D. Brodsky
author: James Howie
published in: Israel Journal of Mathematics
publication date: 1998-02-08T00:00:00Z
review text: Let \(\rho(G)=\bigcap\{N\triangleleft G\), \(G/N\) torsion free\} be the torsion-free radical of a group \(G\), \(\widehat G=G/\rho(G)\) be the torsion-free image of \(G\). Theorem 1.4. Let \(G\) be a group with a presentation having at most 5 relators, each of length at most 3. Then \(\widehat G\) is locally indicable. Theorem 1.6. Let \(A\), \(B\) be torsion-free groups, and \(w\in A*B\) a cyclically reduced word of length at least 2. Suppose \(a\in A\) can be expressed in the form \(a=\prod^n_{i=1}v_iw^{\varepsilon(i)}v_i^{-1}\) with \(n\leq 4\) and \(\varepsilon(i)=\pm1\) for each \(i\). Then \(a=1\). F. Levin's conjecture (1962) is equivalent to the assertion of this theorem, without the restriction on \(n\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1033786
zbMATH Keywords: locally indicable groups
zbMATH Keywords: equations over groups
zbMATH Keywords: torsion-free radical
zbMATH Keywords: presentations
zbMATH Keywords: relators
zbMATH Keywords: torsion-free groups
zbMATH Keywords: cyclically reduced words
reviewed by: Leonid A. Bokut'
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1033786