#### Volume 9, issue 1 (2005)

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Distances of Heegaard splittings

### Aaron Abrams and Saul Schleimer

Geometry & Topology 9 (2005) 95–119
 arXiv: math.GT/0306071
##### Abstract

J Hempel showed that the set of distances of the Heegaard splittings $\left(S,\mathsc{V},{h}^{n}\left(\mathsc{V}\right)\right)$ is unbounded, as long as the stable and unstable laminations of $h$ avoid the closure of $\mathsc{V}\subset \mathsc{P}\mathsc{ℳ}\mathsc{ℒ}\left(S\right)$. Here $h$ is a pseudo-Anosov homeomorphism of a surface $S$ while $\mathsc{V}$ 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 $\left(S,\mathsc{V},{h}^{n}\left(\mathsc{V}\right)\right)$ 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
Primary: 57M99
Secondary: 51F99