Let and
be subgroups of a
free group of ranks
and ,
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 hyperbolic
–manifolds.
We also prove the following particular case of the Hanna
Neumann Conjecture, which has also been obtained by Louder. If
has rank at
least , then
has rank no
more than .
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