Abstract
J Hempel showed that the set of distances of the Heegaard splittings is unbounded, as long as the stable and unstable laminations of avoid the closure of . Here is a pseudo-Anosov homeomorphism of a surface while is the set of isotopy classes of simple closed curves in bounding essential disks in a fixed handlebody.
With the same hypothesis we show the distance of the splitting grows linearly with , answering a question of A Casson. In addition we prove the converse of Hempel’s theorem. Our method is to study the action of on the curve complex associated to . We rely heavily on the result, due to H Masur and Y Minsky, that the curve complex is Gromov hyperbolic.
Citation
Aaron Abrams. Saul Schleimer. "Distances of Heegaard splittings." Geom. Topol. 9 (1) 95 - 119, 2005. https://doi.org/10.2140/gt.2005.9.95
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