Word length statistics for Teichm\"uller geodesics and singularity of harmonic measure

TL;DR

This paper investigates the geometric properties of Teichmüller geodesics under different boundary measures, revealing that harmonic measure geodesics exhibit distinct metric behavior and are singular relative to Lebesgue measure.

Contribution

It establishes the singularity of harmonic measure for Teichmüller geodesics and analyzes the ratio of word to relative metrics along these geodesics, providing new insights into their geometric and measure-theoretic properties.

Findings

Ratio tends to infinity for Lebesgue measure geodesics

Ratio remains finite for harmonic measure geodesics

Harmonic measure is singular with respect to Lebesgue measure

Abstract

Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the ratio between the word metric and the relative metric of approximating mapping class group elements along a geodesic ray, and prove that this ratio tends to infinity along almost all geodesics with respect to Lebesgue measure, while the limit is finite along almost all geodesics with respect to harmonic measure. As a corollary, we establish singularity of harmonic measure. We show the same result for cofinite volume Fuchsian groups with cusps. As an application, we answer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunov expansion exponent.