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We study the Furstenberg-entropy realization problem for stationary actions.It is shown that for finitely supported probability measures on free groups,any a-priori possible entropy value can be realized as the entropy of anergodic stationary action. This generalizes results of Bowen. The stationaryactions we construct arise via invariant random subgroups (IRSs), based onideas of Bowen and Kaimanovich. We provide a general framework for constructinga continuum of ergodic IRSs for a discrete group under some algebraicconditions, which gives a continuum of entropy values. Our tools apply forexample, for certain extensions of the group of finitely supported permutationsand lamplighter groups, hence establishing full realization results for thesegroups. For the free group, we construct the IRSs via a geometric constructionof subgroups, by describing their Schreier graphs. The analysis of the entropyof these spaces is obtained by studying the random walk on the appropriateSchreier graphs.

Furstenberg Entropy of Intersectional Invariant Random Subgroups