On translation lengths of pseudo-Anosov maps on curve graph

Abstract

We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. In the general case of surfaces, 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. Four main applications of our theorem are the following:: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov and pseudo-Anosov map on the curve graph to that on Teichmüller space and to give a new class of pseudo-Anosov optimizing ratio, (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length and (d) to give a partial answer of how much power is needed for Dehn twists to generate right-angled Artin subgroup of the mapping class group.