Random walks on cocompact Fuchsian and Kleinian groups

TL;DR

This paper proves the conjecture that the hitting measure of certain random walks on most cocompact Fuchsian and Kleinian groups is singular at infinity, using algebraic, geometric convergence, and hyperbolic Dehn filling techniques.

Contribution

It establishes the singularity of hitting measures for a broad class of random walks on cocompact Fuchsian and Kleinian groups, confirming a conjecture from 2011.

Findings

Hitting measures are singular at infinity for most cocompact Fuchsian and Kleinian groups.

The proof employs algebraic and geometric convergence methods.

Hyperbolic Dehn filling is used to analyze the measures.

Abstract

The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $\Gamma$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups.