An estimate for the rank of the intersection of subgroups in free amalgamated products of two groups with normal finite amalgamated subgroup. (Q2842989)

The Hanna Neumann inequality \(\overline r(H_1\cap H_2)\leq 2\overline r(H_1)\overline r(H_2)\) bounds the reduced rank of the intersection of two finitely generated subgroups \(H_1,H_2\) of a free group \(G\). (The reduced rank \(\overline r(H)=\max(0,r(H)-1)\) where \(r(H)\) is the rank of \(H\).) \textit{W. Dicks} and \textit{S. V. Ivanov} [Math. Proc. Camb. Philos. Soc. 144, No. 3, 511-534 (2008; Zbl 1154.20025)] generalized it to the case when \(G\) is a free product of groups.NEWLINENEWLINE This article generalizes the latter result to the case when \(G=G_1*_TG_2\) is an amalgamated free product with finite, normal, amalgamated subgroup \(T\) and \(H_1,H_2\) are factor-free subgroups (that is, they intersect trivially with the conjugates to the factors \(G_1,G_2\)). Two upper bounds are given. The weaker bound of \(6|T|\overline r(H_1)\overline r(H_2)\) is easier to state, but the stronger bound is sharp.NEWLINENEWLINE The bounds are obtained quickly with an argument that makes use of Ivanov and Dicks' bounds for free products (those bounds correspond directly to the case \(T=1\)). The remainder of the paper proves that the stronger bound is sharp, using a graph-theoretic approach due to Ivanov.