The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups

TL;DR

This paper establishes a formula linking the Hausdorff dimension of harmonic measures on boundaries of relatively hyperbolic groups to entropy and drift, revealing geometric properties and metric classifications of these boundaries.

Contribution

It provides a novel dimension formula for harmonic measures on various boundaries of relatively hyperbolic groups, connecting entropy, drift, and growth rates, and characterizes doubling visual metrics.

Findings

Dimension equals entropy over drift for harmonic measures.

Visual metric dimension matches the group's growth rate.

Doubling metrics occur only for virtually free groups.

Abstract

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