Magnus intersections in one-relator products. (Q2491067)

Let \(G=(*_{\lambda\in\Lambda}A_\lambda)/\langle\langle R\rangle\rangle\) be an one-relator product of locally indicable groups. For a subset \(M\) of \(\Lambda\), \textit{S.~D.~Brodskii} proved [in Sib. Math. J. 25, 235--251 (1984); translation from Sib. Mat. Zh. 25, No. 2(144), 84--103 (1984; Zbl 0579.20020)] that, under some conditions, the group \(A_M=*_{\mu\in M}A_\mu\) embeds into \(G\). Let \(M,N\subseteq\Lambda\). The question is: under what conditions \(A_M\cap A_N=A_{M\cap N}\)? \textit{D.~J.~Collins} [in Lond. Math. Soc. Lect. Note Ser. 311, 255-296 (2004; Zbl 1078.20031)] has proved that if \(G=(*_{\lambda\in\Lambda}A_\lambda)/\langle\langle R\rangle\rangle\) is a one-relator group, then \(A_M\cap A_N=A_{M\cap N}*I\), where \(I\) is a free group of rank 0 or 1. Here it is obtained a generalization of Collins' result to the case of arbitrary locally indicable factors. This is obtained proving first that if the relator in a one-relator product of locally indicable groups is a proper power then \(A_M\cap A_N=A_{M\cap N}\) (a generalization of a result of \textit{B.~B.~Newman} [in Bull. Am. Math. Soc. 74, 568--571 (1968; Zbl 0174.04603)]). In the case where \(A_{M\cap N}\) is strictly contained in \(A_M\cap A_N\) it is described in detail the relator \(R\) and the subgroup \(I\) such that \(A_M\cap A_N=A_{M\cap N}*I\). The result that the subgroup \(I\) is a free group of rank 0 or 1 is derived by appeal to a Theorem of \textit{S.~Brodskii} [in Algebraic systems, Interuniv. Collect. sci. Works, Ivanovo 1981, 51--77 (1981; Zbl 0525.20017)]. These results follow from a systematic study of `minimal intersection van Kampen diagrams'. Moreover, this study enables the author to give an algorithm to decide, in the case of one-relator groups, whether or not the subgroup \(I\) in \(A_M\cap A_N=A_{M\cap N}*I\) is trivial or infinite cyclic, and in the last case to give a generator of \(I\).