Paper published by Geometry and Topology

From: Geometry and Topology Journal (gt_at_maths.warwick.ac.uk)
Date: 12/23/04 

Date: Thu, 23 Dec 2004 00:30:04 +0000 (UTC)

The following paper has been published:

Geometry and Topology, Volume 9 (2005) Paper no. 2, pages 95--119

URL:
http://www.maths.warwick.ac.uk/gt/GTVol9/paper2.abs.html

Title:
Distances of Heegaard splittings

Author(s):
Aaron Abrams, Saul Schleimer

Abstract:

J Hempel [Topology, 2001] showed that the set of distances of the
Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable
and unstable laminations of h avoid the closure of V in PML(S). Here h
is a pseudo-Anosov homeomorphism of a surface S while V is the set of
isotopy classes of simple closed curves in S bounding essential disks
in a fixed handlebody.

With the same hypothesis we show the distance of the splitting (S,V,
h^n(V)) grows linearly with n, answering a question of A Casson. In
addition we prove the converse of Hempel's theorem. Our method is to
study the action of h on the curve complex associated to S. We rely
heavily on the result, due to H Masur and Y Minsky
[Invent. Math. 1999], that the curve complex is Gromov hyperbolic.

AMS Classification Numbers. Primary: 57M99
Secondary: 51F99

Keywords:
Curve complex, Gromov hyperbolicity, Heegaard splitting

Received: 5 June 2003
Revised: 20 December 2004
Accepted: 29 September 2004
Published: 22 December 2004

Proposed: Martin Bridson
Seconded: Cameron Gordon, Joan Birman

Author(s) address(es):
Department of Mathematics, Emory University
Atlanta, Georgia 30322, USA
and
Department of Mathematics, Rutgers University
Piscataway, New Jersey 08854, USA

Email: abrams@mathcs.emory.edu, saulsch@math.rutgers.edu
URL: http://www.mathcs.emory.edu/~abrams, http://www.math.rutgers.edu/~saulsch