Aaron Abrams and Saul Schleimer

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 splitting (S,V , hn(V)) 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 [10], that the curve complex is Gromov hyperbolic.