The famous Hanna Neumann Conjecture (now the Friedman--Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to 'quantify' this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary H, K, and about my recent results in this direction. In particular, we resolve the remaining open case (m=4) of Guzman's `Group-Theoretic Conjecture' in the affirmative. This has some interesting corollaries for the geometry of hyperbolic 3-manifolds. Our methods rely on recasting the topological pushout of core graphs in terms of the Dicks graphs introduced in the context of his Amalgamated Graph Conjecture. This allows to translate the question of existence of a pair of subgroups H,K with prescribed ranks of joins and intersections into graph theoretic language, and completely resolve it in some cases. In particular, we completely describe the locus of realizable values of ranks in the case when the rank of one of the subgroups H,K equals two.[-]

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.[-]