Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

TL;DR

This paper proves that for certain infinite covolume subgroups of higher rank semisimple Lie groups, the associated Furstenberg boundary measure is singular, contrasting with the lattice case and extending understanding of boundary measures.

Contribution

It establishes the singularity of the Furstenberg measure for infinite covolume, Zariski-dense subgroups in higher rank semisimple Lie groups, a novel result in this context.

Findings

Furstenberg boundary measure is singular for infinite covolume subgroups.

Contrasts with the lattice case where measures are not singular.

First such result for higher rank semisimple Lie groups.

Abstract

We consider symmetric random walks on discrete, Zariski-dense subgroups $\Gamma$ of a semisimple Lie group $G$ with Property (T). We prove that if $\Gamma$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank.