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,hⁿ(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V ⊂ 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,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

Primary: 57M99

Secondary: 51F99

References

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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