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Let μ\mu be a probability measure on Γ\Gamma with finite entropy, such that its help generates Γ\Gamma as a semigroup. Yet, this is sufficient to define a boundary map, therefore it is sensible to ask whether the resulting topological boundary is the Poisson boundary. On the opposite hand, if one drops the second condition, the random walk nonetheless converges, but in a weaker sense, i.e. within the Furstenberg-Satake compactification, whose boundary is G/PG/P GR85. In this paper, we drop each moment situation, and establish the Poisson boundary for random walks on discrete subgroups of semisimple Lie groups assuming simply that the entropy is finite.
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- Theorem 1.1 is the higher rank analog of the primary theorem of CFFT25, that for hyperbolic groups, and extra generally acylindrically hyperbolic groups, set up the identification of the Poisson boundary with the Gromov boundary underneath finite entropy and with out second conditions.
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