Singular measures on the limit set of a Kleinian group

TL;DR

This paper proves that for certain classes of Kleinian groups, the Patterson-Sullivan measure is singular relative to the harmonic measure derived from a symmetric random walk, highlighting differences in measure behavior.

Contribution

It establishes the singularity of Patterson-Sullivan measures compared to harmonic measures for specific types of Kleinian groups, extending understanding of measure properties in hyperbolic geometry.

Findings

Patterson-Sullivan measure is singular with respect to harmonic measure for geometrically infinite groups without parabolics.

Singularity also holds for Gromov hyperbolic groups with parabolics.

Results apply to finitely generated torsion-free Kleinian groups under specified conditions.

Abstract

We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $\mu$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan measure is singular with respect to the harmonic measure coming from $\mu$.