Algebraic & Geometric Topology 9 (2009) 305–325

DOI: 10.2140/agt.2009.9.305

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:

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.

Keywords

free group, rank, intersection, join, Hanna Neumann Conjecture

Mathematical Subject Classification

Primary: 20E05

Secondary: 57M50

References

Publication

Received: 31 January 2008
Revised: 18 August 2008
Accepted: 28 January 2009
Published: 23 February 2009

Authors