The rank three case of the Hanna Neumann Conjecture

For a free group G, rk(G) denotes the rank of G, and, for each positive integer n, rk-n(G) denotes max{rk(G) - n, 0}. Let H and K be finitely generated subgroups of a free group. Hanna Neumann conjectured that rk-1(H ∩ K) ≤ rk-1(H) rk-1(K). We prove that rk-1(H ∩ K) ≤ rk-1(H) rk-1(K) + rk-3(H) rk-3(K). This extends results of Hanna Neumann, R. G. Burns and G. Tardos, and shows that, if H has rank three or less, then the conjectured inequality holds. Our argument consists of proving the corresponding case of the Amalgamated Graph Conjecture, and therefore applies to Walter Neumann's strengthened version of the Hanna Neumann Conjecture.