Comptage probabiliste sur la fronti\`ere de Furstenberg
Richard Aoun

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 be a real linear semisimple algebraic group without compact factors and a Zariski dense subgroup of . In this paper, we use a probabilistic counting in order to study the asymptotic properties of acting on the Furstenberg boundary of . First, we show that the components of the elements of in the KAK decomposition of 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 . 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 will eventually generate a free subgroup. This result…
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Taxonomy
Topicsadvanced mathematical theories · Mathematical Dynamics and Fractals · Advanced Topology and Set Theory
