The Hanna Neumann conjecture states that the intersection of two nontrivial subgroups of rank k + 1 and l + 1 of a free group has rank at most kl + 1. In a recent paper  W. Dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture. He used this equivalence to reprove all known upper bounds on the rank of the intersection. We use his method to improve these bounds. In particular we prove an upper bound of 2kl - k - l + 1 for the rank of the intersection above (k, l ≧ 2) improving the earlier 2kl - min(k, l) bound of . We prove a special case of the amalgamated graph conjecture in the hope that it may lead to a proof of the general case and thus of the strengthened Hanna Neumann conjecture.
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