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.

Notes on file formats

Aaron Abrams, Saul Schleimer
Department of Mathematics, Emory University
Atlanta, Georgia 30322, USA
and
Department of Mathematics, Rutgers University
Piscataway, New Jersey 08854, USA
Email: abrams@mathcs.emory.edu, saulsch@math.rutgers.edu
URL: http://www.mathcs.emory.edu/~abrams and http://www.math.rutgers.edu/~saulsch

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