Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε > 0 such that the Furstenberg entropy of any (G, µ)-stationary ergodic space is either zero or larger than ε.Virtually free groups, such as SL 2 (Z), do not have property (T), and neither do their extensions, such as surface groups. For these, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies between zero and the Poisson boundary entropy.

A countable discrete group G is called Choquet-Deny if for every non-degenerate probability measure µ on G it holds that all bounded µ-harmonic functions are constant. We show that a finitely generated group G is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that G is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when G is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

Let (G,mu) be a discrete group equipped with a generating probability measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G, starting from the identity, returns to Gamma with probability one. Let theta be the hitting measure, or the distribution of the position in which the random walk first hits Gamma. We prove that the Furstenberg entropy of a (G,mu)-stationary space, with respect to the induced action of (Gamma,theta), is equal to the Furstenberg entropy with respect to the action of (G,mu), times the index of Gamma in G. The index is shown to be equal to the expected return time to Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu), we prove that the random walk entropy of (Gamma,theta) is equal to the random walk entropy of (G,mu), times the index of Gamma in G.Comment: 16 page

Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and NevoZimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.

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