Random walk speed is a proper function on Teichmüller space

Abstract

Consider a closed surface M with negative Euler characteristic, and an admissible probability measure on the fundamental group of M with a finite first moment. Corresponding to each point in the Teichmüller space of M , there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of M , and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gouëzel’s pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.