Generating systems of groups and Reidemeister-Whitehead torsion
type: Item
instance of: scholarly article
title: Generating systems of groups and Reidemeister-Whitehead torsion
zbMATH Open document ID: 0816.20031
author: Martin Lustig
author: Yoav Moriah
published in: Journal of Algebra
publication date: 1995-04-03T00:00:00Z
review text: Two (minimal) systems of generators of a group are called Nielsen equivalent, if the change of generators can be lifted to the free groups generated by them. The purpose of the paper is to define a generally applicable invariant which can be used to show that a pair of generating systems is not Nielsen equivalent. For a finitely generated group \(G\) an abelian group \({\mathcal N}(G)\) is defined and to a pair \(x\), \(y\) of minimal generating systems an element \({\mathcal N}(y,x) \in {\mathcal N}(G)\) is assigned which only depends on the Nielsen classes of \(x\) and \(y\), and which vanishes, if \(x\) and \(y\) are Nielsen equivalent. \({\mathcal N}(G)\) is the Whitehead group of the quotient of \(\mathbb{Z} G\) modulo the ``Fox ideal'' which is generated by the Fox derivatives of the defining relators of \(G\) by the generators \(x\) of \(G\). To make the invariant computable a homomorphism of \(\mathbb{Z} G\) into a matrix ring over a unitary commutative ring is employed, and the determinant of the image of the Jacobian of the two generating systems \(x\) and \(y\) is considered. If this determinant is not of a certain ``trivial-unit-type'', then \(x\) and \(y\) are not Nielsen equivalent. The authors prove functorial properties of \(\mathcal N\), and consider the behaviour of their invariant under free products. The last paragraph is devoted to applications: finite abelian groups are considered, and known results recovered and extended; knot groups, especially of 2- and certain 3-bridge knots, are treated. These examples can be used to exhibit Nielsen inequivalent generators for 3-manifolds obtained by Dehn surgery on \(S^ 3\) along such knots. This implies non-isotopy of associated Heegaard splittings of this manifold.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 202989
zbMATH Keywords: Nielsen equivalence
zbMATH Keywords: Fox calculus
zbMATH Keywords: torsion
zbMATH Keywords: systems of generators
zbMATH Keywords: change of generators
zbMATH Keywords: free groups
zbMATH Keywords: finitely generated group
zbMATH Keywords: minimal generating systems
zbMATH Keywords: Nielsen classes
zbMATH Keywords: Whitehead group
zbMATH Keywords: Fox derivatives
zbMATH Keywords: relators
zbMATH Keywords: generators
zbMATH Keywords: free products
zbMATH Keywords: finite abelian groups
zbMATH Keywords: knot groups
zbMATH Keywords: Nielsen inequivalent generators
zbMATH Keywords: 3-manifolds
zbMATH Keywords: Dehn surgery
zbMATH Keywords: Heegaard splittings
Wikidata QID: Q63443946
reviewed by: Gerhard Burde
MaRDI profile type: Publication
full work available at URL: jabr.1993.1096
OpenAlex ID: W1998946954
DOI: 10.1006/JABR.1993.1096
description: scientific article; zbMATH DE number 202989