Effective Mapping Class Group Dynamics II: Geometric Intersection Numbers

Abstract

We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the boundary at infinity of Teichmüller space, an effective estimate comparing the Teichmüller and Thurston metrics along mapping class group orbits of Teichmüller space, and, in the sequel and forthcoming work of Honaryar, effective estimates for countings of closed geodesics on closed, negatively curved surfaces. Furthermore, in forthcoming work of Arana-Herrera and Honaryar, the main result of this paper is applied to study the arithmetic/homological complexity of long simple closed geodesics on negatively curved surfaces.