Volume 9, issue 1 (2005)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24, 1 issue

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Distances of Heegaard splittings

Aaron Abrams and Saul Schleimer

Geometry & Topology 9 (2005) 95–119

arXiv: math.GT/0306071


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.

curve complex, Gromov hyperbolicity, Heegaard splitting
Mathematical Subject Classification 2000
Primary: 57M99
Secondary: 51F99
Forward citations
Received: 5 June 2003
Revised: 20 December 2004
Accepted: 29 September 2004
Published: 22 December 2004
Proposed: Martin Bridson
Seconded: Cameron Gordon, Joan Birman
Aaron Abrams
Department of Mathematics
Emory University
Georgia 30322
Saul Schleimer
Department of Mathematics
Rutgers University
New Jersey 08854