#### Volume 9, issue 1 (2009)

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Intersections and joins of free groups

### Richard Peabody Kent IV

Algebraic & Geometric Topology 9 (2009) 305–325
##### 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:

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

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 $\left(h-1\right)\left(k-1\right)+1$.

##### Keywords
free group, rank, intersection, join, Hanna Neumann Conjecture
Primary: 20E05
Secondary: 57M50
##### Publication
Revised: 18 August 2008
Accepted: 28 January 2009
Published: 23 February 2009
##### Authors
 Richard Peabody Kent IV Department of Mathematics Brown University Providence, RI 02912