Distances of Heegaard splittings

Abrams, Aaron¹ ; Schleimer, Saul²

¹ Department of Mathematics, Emory University, Atlanta, Georgia 30322, USA
² Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854, USA

Geometry & topology, Tome 9 (2005) no. 1, pp. 95-119.

Résumé

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.

Détail

DOI : 10.2140/gt.2005.9.95

Keywords: curve complex, Gromov hyperbolicity, Heegaard splitting

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