Volume 9, issue 1 (2009)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 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.

Keywords
free group, rank, intersection, join, Hanna Neumann Conjecture
Mathematical Subject Classification 2000
Primary: 20E05
Secondary: 57M50
References
Publication
Received: 31 January 2008
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