Harmonic measures for distributions with finite support on the mapping class group are singular

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into Teichmüller space converges almost surely to a point in the space PMFPMF of projective measured foliations on the surface. This defines a harmonic measure on PMFPMF. Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure class on PMFPMF