The identity problem for graph products of groups
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.
scientific article; zbMATH DE number 554202