Distances of Heegaard splittings

Abstract

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