Intersections and joins of free groups

Abstract

Let $H$ and $K$ be subgroups of a free group of ranks $h$ and $k\ge h$, respectively. We prove the following strong form of Burns’ inequality:

$$rank\left(H\cap K\right)-1\le 2\left(h-1\right)\left(k-1\right)-\left(h-1\right)\left(\right.rank\left(H\vee K\right)-1\left)\right..$$

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 $3$–manifolds.

We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If $H\vee K$ has rank at least $\left(h+k+1\right)∕2$, then $H\cap K$ has rank no more than $\left(h-1\right)\left(k-1\right)+1$.