Harmonic measures versus quasiconformal measures for hyperbolic groups

Abstract

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.

Besides minor modifications, we provide a new proof that the harmonic measure of a finitely supported random walk on a Fuchsian group with cusps is singular. 52 pp