Intersections and joins of free groups

Let F be a free group. If H and K are subgroups of F , we let H ∨K = 〈H,K〉 denote the join of H and K. We study the relationship between the rank of H ∩K and that of H ∨K for a pair of finitely generated subgroups H and K of F . In particular, we have the following particular case of the Hanna Neumann Conjecture, which has also been obtained by L. Louder [6] using his machinery for folding graphs of spaces [7, 8, 9]. For detailed discussions of the Hanna Neumann Conjecture, see [11, 12, 13, 16, 4, 3]. Theorem 1 (Kent, Louder). Let H and K be nontrivial finitely generated subgroups of a free group of ranks h and k, respectively. If rank(H ∨K)−1≥ h+ k−1 2 then