Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture
Let \(G\) be a free group. The reduced rank of \(G\) is defined as \(\widetilde r(G)= \max\{\text{rank}(G)- 1,0\}.\) \textit{W. D. Neumann} [Lect. Notes Math. 1456, 161-170 (1990; Zbl 0722.20016)] asked if the following statement is true: If \(H\) and \(K\) are subgroups of \(G\) then \[ \sum_{HgK\in H\backslash G/K}\widetilde r(H^ g\cap K)\leq \widetilde r(H) \widetilde r(K). \] This statement is called the strengthened Hanna Neumann conjecture because it is a stronger version of a still unsolved conjecture made by \textit{Hanna Neumann} [Publ. Math. 4, 186-189 (1956; Zbl 0070.02001)]. Here \(H^ g\) denotes \(g^{-1} Hg\), and \(H\backslash G/K\) denotes the set of double cosets \(HgK\), \(g\in G\). In this article it is shown that this new conjecture is equivalent to the following previously uncontemplated conjecture, called the amalgamated graph conjecture: If \(\Delta\) is any finite bipartite graph for which there exist three finite bipartite \(\Delta\)-graphs \(A\), \(B\), \(\Gamma\) such that the finite bipartite graph \((A\vee_ \Delta B)\vee (B\vee_ \Delta \Gamma)\vee (\Gamma\vee_ \Delta A)\) is simple-edged and is the disjoint union of two isomorphic bipartite subgraphs, then \(\Delta\) has at most half as many edges as the complete bipartite graph with the same vertex set. Terminology used here is as follows: a bipartite graph has each vertex coloured black or white, and each edge joins two vertices of different colours; a \(\Delta\)-graph is a graph given with a specified copy of \(\Delta\) as a bipartite subgraph; simple-edged means two vertices are joined by at most one edge, \(\vee\) denotes disjoint union, and \(\vee_ \Delta\) the pushout identifying the two copies of \(\Delta\). Simple proofs of some previously known results are given to demonstrate the strength of this equivalence.
scientific article; zbMATH DE number 703854