Distances of Heegaard splittings

Abrams, Aaron; Schleimer, Saul

doi:10.2140/gt.2005.9.95

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Distances of Heegaard splittings

Aaron Abrams and Saul Schleimer

Geometry & Topology 9 (2005) 95–119

DOI: 10.2140/gt.2005.9.95

arXiv: math.GT/0306071

Abstract

J Hempel 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 $V \subset P ℳ ℒ (S)$ . 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, 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/

Volume 9, issue 1 (2005)