On the rank of intersection of subgroups of a free product of groups
The authors study the intersection of two subgroups of a free product of two groups. It is well-known that the intersection of two finitely generated subgroups of a free group is again finitely generated and there is a bound of its rank in terms of the ranks of these subgroups, given by \textit{A. G. Howson} in 1954 [J. Lond. Math. Soc. 29, 428-434 (1954; Zbl 0056.02106)]. This was improved in 1958 by H. Neumann. This property is called the Howson property. In case of a free product of groups one has the Kurosh rank \(\text{Krk }H\) of a subgroup \(H\), as the number of free factors arising from the Kurosh subgroup theorem of free products. So the authors extend the above-mentioned result of free groups in case of subgroups of a free product of two groups \(A\) and \(B\). Their main result states that: If \(H,K\leq A*B\) have finite Kurosh ranks, then \[ \text{KrK}(H\cap K)-1\leq 2(\text{Krk }H-1)(\text{Krk }K-1)+2\min\{\text{Krk }H-1,\text{Krk }K-1\}. \] This improves a similar result of \textit{T. Soma} [J. Pure Appl. Algebra 66, No. 1, 81-95 (1990; Zbl 0713.20020)]. The proof is geometric. The authors first restate the Kurosh subgroup theorem for the free product \(A*B\) and then define the ``Kurosh graph'' of a subgroup \(H\) of \(A*B\) arising actually from the realization of \(H\) as the fundamental group of a certain surface complex.
scientific article; zbMATH DE number 1121645
scientific article