Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Abstract

Given a measure on the Thurston boundary of Teichmüller 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 word length of approximating mapping class group elements along a geodesic ray, and prove that this quantity grows superlinearly in time along almost all geodesics with respect to Lebesgue measure, while along almost all geodesics with respect to harmonic measure the growth is linear. As a corollary, the harmonic and Lebesgue measures are mutually singular. We also prove a similar result for the ratio between the word metric and the relative metric (i.e. the induced metric on the curve complex).