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 Teichm\"uller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichm\"uller 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 Teichm\"uller geodesic rays to the boundary at infinity of Teichm\"uller space, an effective estimate comparing the Teichm\"uller and Thurston metrics along mapping class group orbits of Teichm\"uller space, and, in the sequel, effective estimates for countings of filling closed geodesics on closed, negatively curved surfaces.