Groups, graphs, and the Hanna Neumann conjecture. (Q2885378)

Let \(F\) be a free group. W. Neumann proposed the following Strengthened Hanna Neumann Conjecture (SHNC). Let \(A\backslash F/B\) be the set of all double cosets \(AgB\) for \(g\in F\) and \(s\colon A\backslash F/B\to F\) be a section of the quotient map \(F\to A\backslash F/B\):NEWLINENEWLINE Conjecture (SHNC). Suppose \(F\) is a free group and \(A\) and \(B\) are finitely generated subgroups. Then \(\sum_{u\in s(A\backslash F/B)}\overline r(A^u\cap B)\leq\overline r(A)\cdot\overline r(B)\).NEWLINENEWLINE Here \(C^u\) denotes \(u^{-1}Cu\), and \(\overline r(C)\) denotes the reduced rank of \(C\), i.e., the number \(\max\{0,\text{rk\,}C-1\}\).NEWLINENEWLINE The purpose of this note is to write a proof of the SHNC in terms of groups and graphs. Also explicit examples are given showing that the upper bound in SHNC is sharp.