Realizable ranks of joins and intersections of subgroups in free groups

Abstract

The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups [Formula: see text] and [Formula: see text] of a non-abelian free group. It is an interesting question to “quantify” this bound with respect to the rank of [Formula: see text], the subgroup generated by [Formula: see text] and [Formula: see text]. We describe a set of realizable values [Formula: see text] for arbitrary [Formula: see text], [Formula: see text], and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for [Formula: see text] and [Formula: see text] with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of [Formula: see text], [Formula: see text] are not realizable, thus resolving the remaining open case [Formula: see text] of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic 3-manifolds with a 6-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when [Formula: see text].