Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups

TL;DR

This paper constructs a random walk on groups hyperbolic relative to virtually nilpotent subgroups, showing the harmonic measure aligns with the Bowditch boundary and exhibits exact dimensionality and Ahlfors-regularity.

Contribution

It introduces a new random walk model with harmonic measures matching the Bowditch boundary, linking Green metrics, conformal densities, and geometric properties.

Findings

Harmonic measure is a conformal density on the Bowditch boundary.

The boundary with visual distance is Ahlfors-regular.

Dimension relates to drift, Green drift, and entropy.

Abstract

For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped graph associated to the group, we construct a random walk whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is exact dimensional on the Bowditch boundary. The latter equipped with a visual distance induced by the Green metric is an Ahlfors-regular metric measure space. The dimension is given in terms of the drift, a Green drift and the asymptotic entropy. The Patterson-Sullivan density for the action on the cusped graph in this case is doubling, its dimension is obtained by looking at cusp excursions of geodesics.