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- Progress conditions for freeness of the Furstenberg boundary action (2026)
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Let μ\mu be a probability measure on Γ\Gamma with finite entropy, such that its assist generates Γ\Gamma as a semigroup. Yet, this is enough to outline a boundary map, hence it is sensible to ask whether the ensuing topological boundary is the Poisson boundary. On the opposite hand, if one drops the second condition, the random stroll 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 second condition, and identify the Poisson boundary for random walks on discrete subgroups of semisimple Lie groups assuming just that the entropy is finite.
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