The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups

The paper studies the Hausdorff dimension of harmonic measures on various boundaries a relatively hyperbolic group which are associated with random walks driven by probability measure finite first moment. With respect to Floyd metric and shortcut metric, we prove that equals ratio entropy drift walk. If is infinitely-ended, same formula obtained for end boundary endowed visual metric. In addition, identified growth rate word These results complemented characterization doubling metrics accessible infinitely-ended groups: if only virtually free. Consequently, there at least two different bi-Hölder classes (and thus quasi-symmetric classes) boundary.