## Algebraic & Geometric Topology

### Intersections and joins of free groups

Richard Peabody Kent

#### Abstract

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:

$rank ( H ∩ K ) − 1 ≤ 2 ( h − 1 ) ( k − 1 ) − ( h − 1 ) rank ( H ∨ K ) − 1 .$

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 $H∨K$ has rank at least $(h+k+1)∕2$, then $H∩K$ has rank no more than $(h−1)(k−1)+1$.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 1 (2009), 305-325.

Dates
Revised: 18 August 2008
Accepted: 28 January 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796967

Digital Object Identifier
doi:10.2140/agt.2009.9.305

Mathematical Reviews number (MathSciNet)
MR2482079

Zentralblatt MATH identifier
1170.20017

Subjects
Primary: 20E05: Free nonabelian groups
Secondary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

Kent, Richard Peabody. Intersections and joins of free groups. Algebr. Geom. Topol. 9 (2009), no. 1, 305--325. doi:10.2140/agt.2009.9.305. https://projecteuclid.org/euclid.agt/1513796967

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