The identity problem for graph products of groups
type: Item
instance of: scholarly article
title: The identity problem for graph products of groups
zbMATH Open document ID: 0804.20026
author: Young-Gheel Baik
author: James Howie
author: Stephen J. Pride
published in: Journal of Algebra
publication date: 1995-01-12T00:00:00Z
review text: Let \(\Gamma\) be a graph with vertex set \(V\) and edge set \(E\) with chosen orientation \(E^ +\). For each \(v\in V\) let \(G_ v\) be a group given by a presentation \({\mathcal P}_ v=\langle A_ v\mid S_ v\rangle\) and for each \(e\in E^ +\) let \(R_ e\) consist of all words \([a,b]= aba^{-1} b^{-1}\) (\(a\in A_ v\), \(b\in A_ u\)) where \(v\) is the initial vertex of \(e\) and \(u\) is the terminal one. The group \(G\) defined by the presentation \({\mathcal G}=\langle A=\bigcup_{v\in V} A_ v\); \(S=\bigcup_{v\in V} S_ v\), \(R=\bigcup_{e\in E^ +} R_ e\rangle\) is called a graph product of the groups \(G_ v\), \(v\in V\). The authors determine a set of defining relations for the relation module of \(G\) (corresponding to \(\mathcal G\)) in terms of its well-known generators. This amounts to determining generators for the module of identities among relators of the presentation \(\mathcal G\). To obtain the result they make use of some standard facts about identity sequences and the technique of spherical pictures.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F06
zbMATH DE Number: 554202
zbMATH Keywords: words
zbMATH Keywords: presentations
zbMATH Keywords: graph products
zbMATH Keywords: relations
zbMATH Keywords: relation modules
zbMATH Keywords: generators
zbMATH Keywords: relators
zbMATH Keywords: identity sequences
zbMATH Keywords: spherical pictures
MaRDI profile type: Publication
full work available at URL: jabr.1993.1247
OpenAlex ID: W2068297231
DOI: 10.1006/JABR.1993.1247
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description: scientific article; zbMATH DE number 554202