#### 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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