Kaimanovich-Masur showed that for a non-elementary finitely generated random walk on the mapping class group of a surface, almost every sample path in the Teichmüller space converges to the boundary. This gives the hitting measure on the boundary of the Teichmüller space, which is shown to be the unique stationary measure. 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 our generalization of Tukia's rigidity theorem for Kleinian groups to a wide class of group actions, which also generalizes Mostow's rigidity. 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 joint work with Andrew Zimmer.