Let H and K be subgroups of a free group of ranks h and k ≥ h, respectively. We
prove the following strong form of Burns’ inequality:
A corollary of this, also obtained by L Louder and D B McReynolds, has been used
by M Culler and P Shalen to obtain information regarding the volumes of
We also prove the following particular case of the Hanna Neumann Conjecture,
which has also been obtained by Louder. If H ∨ K has rank at least (h + k + 1) ∕ 2,
then H ∩ K has rank no more than (h − 1)(k − 1) + 1.