| | |
Summary: DISTANCES OF HEEGAARD SPLITTINGS
AARON ABRAMS AND SAUL SCHLEIMER
Abstract. J. Hempel in [6] showed that the set of distances of
the Heegaard splittings (S, V, hn
(V)) is unbounded, as long as the
stable and unstable laminations of h avoid the closure of V
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 that the distance of the split-
ting (S, V, hn
(V)) grows linearly with n, answering a question of
A. Casson. In addition we prove the converse of Hempel's theo-
rem. 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 [10], that the curve complex is Gromov hyperbolic.
1. Introduction
J. Hempel has recently introduced a new measure of the complexity
of a Heegaard splitting called the distance of the splitting [6]. This
definition is a conscious extension of A. Casson and C. Gordon's notion
|
