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Published November 16, 2016 | Submitted
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An Abramov formula for stationary spaces of discrete groups


Let (G, μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G, μ)-stationary space, with respect to the induced action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G, μ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the Furstenberg-Poisson boundary of (G, μ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G, μ), times the index of Γ in G.

Additional Information

Date: April 25, 2012. Submitted on 24 Apr 2012. Yuri Lima is supported by the European Research Council, grant 239885. Omer Tamuz is supported by ISF grant 1300/08, and is a recipient of the Google Europe Fellowship in Social Computing, and this research is supported in part by this Google Fellowship.

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