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arXiv · 2508.06329

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

Homin Lee·Wouter Van LimBeek·Giulio Tiozzo
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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.

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Homin Lee, Wouter Van LimBeek, Giulio Tiozzo. 2025-08-08. Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank. https://arxiv.org/abs/2508.06329

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