A note on the singularity conjecture for infinite covolume discrete subgroups

TL;DR

This paper investigates the singularity of stationary measures for random walks on semisimple Lie groups with infinite covolume subgroups, revealing conditions under which these measures are singular to Lebesgue measure.

Contribution

It establishes new results on the singularity of stationary measures for random walks on semisimple Lie groups, including cases with property (T) and rank one, without requiring moment or symmetry conditions.

Findings

Stationary measure is singular for groups with property (T).

Singularity holds for rank one groups with finite first moment.

Provides conditions for singularity in general semisimple groups.

Abstract

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a finite first moment, we again show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. For general semisimple Lie groups, we also obtain a sufficient condition for the singularity of the stationary measure and a general Patterson-Sullivan measure.