Comptage probabiliste sur la fronti\`ere de Furstenberg

TL;DR

This paper employs probabilistic counting methods to analyze the asymptotic behavior of Zariski dense subgroups acting on the Furstenberg boundary of real semisimple algebraic groups, establishing independence, dimension positivity, and free subgroup generation.

Contribution

It introduces a probabilistic approach to study the asymptotic properties of group actions on the Furstenberg boundary, providing new proofs and generalizations of existing results.

Findings

Asymptotic independence of K components in the KAK decomposition.

Positivity of Hausdorff dimension of stationary measure.

Probabilistic proof that two random walks generate a free subgroup.

Abstract

Let $G$ be a real linear semisimple algebraic group without compact factors and $\Gamma$ a Zariski dense subgroup of $G$. In this paper, we use a probabilistic counting in order to study the asymptotic properties of $\Gamma$ acting on the Furstenberg boundary of $G$. First, we show that the $K$ components of the elements of $\Gamma$ in the KAK decomposition of $G$ become asymptotically independent. This result is an analog of a result of Gorodnik-Oh in the context of the Archimedean counting. Then, we give a new proof of a result of Guivarc'h concerning the positivity of the Hausdorff dimension of the unique stationary probability measure on the Furstenberg Boundary of $G$. Finally, we show how these results can be combined to give a probabilistic proof of the Tit's alternative; namely that two independent random walks on $\Gamma$ will eventually generate a free subgroup. This result answered a question of Guivarc'h and was published earlier by the author. Since we're working with the field of real numbers, we give here a more direct proof and a more general statement.