On Translation Lengths of Pseudo-Anosov Maps on the Curve Graph

Abstract

We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve graph to that on Teichm\"uller space, (c) giving a partial answer of how much powers will be needed for Dehn twists to generate right-angled Artin subgroup of the mapping class group.