On a theorem of Furstenberg


A theorem of Furstenberg from 1967 states that if \Gamma is a lattice in a semisimple Lie group G, then there exists a measure on \Gamma with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. We will discuss generalizations of this theorem in the setting of the Mapping class group and Gromov hyperbolic groups.