Comptage probabiliste sur la fronti\`ere de Furstenberg

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.