Geometry & Topology, Vol. 9 (2005) Paper no. 2, pages 95--119.

Distances of Heegaard splittings

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.

Keywords. Curve complex, Gromov hyperbolicity, Heegaard splitting

AMS subject classification. Primary: 57M99. Secondary: 51F99.

DOI: 10.2140/gt.2005.9.95

E-print: arXiv:math.GT/0306071

Submitted to GT on 5 June 2003. (Revised 20 December 2004.) Paper accepted 29 September 2004. Paper published 22 December 2004.

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Aaron Abrams, Saul Schleimer
Department of Mathematics, Emory University
Atlanta, Georgia 30322, USA
Department of Mathematics, Rutgers University
Piscataway, New Jersey 08854, USA
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