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

TL;DR

This paper investigates the translation lengths of pseudo-Anosov maps on the curve graph, providing new estimates and applications related to minimal translation words, optimization of translation length ratios, and subgroup generation.

Contribution

It introduces a method to construct pseudo-Anosov maps with geodesic axes and estimates their stable translation lengths, extending understanding of their geometric properties.

Findings

Pseudo-Anosov maps as products of Dehn twists have geodesic axes.

Derived estimates for stable translation lengths with multicurves.

Applications include minimal translation words and subgroup generation insights.

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.