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 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.