Volume 9, issue 1 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Distances of Heegaard splittings

Aaron Abrams and Saul Schleimer

Geometry & Topology 9 (2005) 95–119

arXiv: math.GT/0306071

Abstract

J Hempel showed that the set of distances of the Heegaard splittings (S,V,hn(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of VP(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,hn(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, that the curve complex is Gromov hyperbolic.

Keywords
curve complex, Gromov hyperbolicity, Heegaard splitting
Mathematical Subject Classification 2000
Primary: 57M99
Secondary: 51F99
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Publication
Received: 5 June 2003
Revised: 20 December 2004
Accepted: 29 September 2004
Published: 22 December 2004
Proposed: Martin Bridson
Seconded: Cameron Gordon, Joan Birman
Authors
Aaron Abrams
Department of Mathematics
Emory University
Atlanta
Georgia 30322
USA
Saul Schleimer
Department of Mathematics
Rutgers University
Piscataway
New Jersey 08854
USA
http://www.math.rutgers.edu/~saulsch/