Kaimanovich-Masur showed that for a generic random walk on the mapping class group of a surface, there exists a unique stationary measure on the boundary of the Teichmüller space. They conjectured that the stationary measure is singular to all Patterson-Sullivan (or, conformal) measures for the group generated by the random walk. In this talk, I will present an affirmative answer to this conjecture for a certain class of random walks, showing the singularity with all Patterson-Sullivan measures. The proof is based on a generalization of Mostow's rigidity and Tukia's rigidity to a wide class of group actions. If time permits, we will also discuss analogous singularity results for any finitely generated Kleinian groups and some discrete subgroups of higher rank Lie groups.

This is a joint work with Andrew Zimmer.