Equations over groups, and groups with one defining relation
type: Item
instance of: scholarly article
title: Equations over groups, and groups with one defining relation
zbMATH Open document ID: 0579.20020
DOI: 10.1007/BF00971461
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: This paper contains detailed proofs of important results in combinatorial group theory, at least some of which the author originally obtained as far back as 1979. The first result generalizes Magnus' Freiheitssatz: If A and B are locally indicable groups (i.e. with every nontrivial finitely generated subgroup having an infinite cyclic quotient) and w is any cyclically reduced word in A*B of length \(\geq 2\), then the normal closure of w in A*B intersects A and B trivially. The proof is purely algebraic, along the general lines of Magnus' proof. (A different proof was given independently by \textit{J. Howie} [J. Reine Angew. Math. 324, 165-174 (1981; Zbl 0447.20032)].) The author goes on to construct a nontrivial class \({\mathcal A}\) of locally indicable groups with the property that for each group \(A\in {\mathcal A}\) and each equation over A, there is a group K in \({\mathcal A}\), containing A as a subgroup, in which the equation has a solution. (From this one readily deduces in particular an affirmative answer to the question of Bokut' as to whether there exists a nontrivial group over which every equation has a solution (in the group).) The class \({\mathcal A}\) is quite wide, containing in particular all free groups, and closed under the formation of ''one-relator products'' \(A_ wB\) (the quotient of A*B by the normal closure of w) where A,B\(\in {\mathcal A}\), and w is as above with the additional property that it is not a proper power. It follows that every torsion-free one-relator group is locally indicable (whence in particular appropriate group rings of such groups have no zero-divisors). It is also shown that a group is locally indicable precisely if it contains a subnormal system with torsion-free abelian factors, so that torsion-free one-relator groups have this structure. An example of a one-relator group with a soluble subnormal system (i.e. an ''SN-group''), but without a soluble normal system (i.e. not an ''SI-group''), is given. Several othe…
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 3927189
zbMATH Keywords: equations over groups
zbMATH Keywords: Freiheitssatz
zbMATH Keywords: locally indicable groups
zbMATH Keywords: finitely generated subgroup
zbMATH Keywords: cyclically reduced word
zbMATH Keywords: normal closure
zbMATH Keywords: free groups
zbMATH Keywords: one-relator products
zbMATH Keywords: torsion-free one-relator group
zbMATH Keywords: group rings
zbMATH Keywords: subnormal system with torsion-free abelian factors
zbMATH Keywords: soluble subnormal system
zbMATH Keywords: SN-group
zbMATH Keywords: SI-group
author: Sergei D. Brodsky
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00971461
OpenAlex ID: W2036980258
description: scientific article; zbMATH DE number 3927189