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Given a partition γ\gamma on the space of pattern paths and ω∈Ω\omega\in\Omega, let γω\gamma\omega denote the category that includes ω\omega. We will take as a base point of SS the coset corresponding to KK, and denote it as oo. Theorem 1.1 is the higher rank analog of the main theorem of CFFT25, that for hyperbolic groups, and more generally acylindrically hyperbolic groups, establish the identification of the Poisson boundary with the Gromov boundary beneath finite entropy and with out second circumstances. We note that finite entropy is a needed assumption for our result and never only a technical artifact. We consider random walks of finite entropy on Zariski-dense discrete subgroups of semisimple Lie teams and show that their Poisson boundary is recognized with the Furstenberg boundary of the corresponding symmetric spaces outfitted with the hitting measure.
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Contents
Nonetheless, these standards require some second situation on the measure μ\mu (finite first second for ray approximation, finite logarithmic moment for the strip approximation) to regulate the geometry of the random stroll. Later, Kaimanovich Kai00 extended this identification result, between the Poisson boundary of the discrete group and the Furstenberg boundary of the ambient Lie group, to the more general case that μ\mu has finite entropy and finite logarithmic moment. We do not assume any moment condition on the random stroll and, in contrast with the previous rank 11 proof, we do not rely on pivoting concept.
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