Intersections and joins of free groups

Kent, Richard Peabody

doi:10.2140/agt.2009.9.305

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Intersections and joins of free groups

Richard Peabody Kent IV

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 $k \geq h$ , respectively. We prove the following strong form of Burns’ inequality:

rank (H \cap K) - 1 \leq 2 (h - 1) (k - 1) - (h - 1) (rank (H \lor 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 $H \lor K$ has rank at least $(h + k + 1) ∕ 2$ , then $H \cap K$ 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

Volume 9, issue 1 (2009)