Furstenberg entropy realizations for virtually free groups and lamplighter groups

Let (G,µ) be a discrete group with a generating probability measure. Nevo showed that if G has Kazhdan’s property (T), then there exists ɛ > 0 such that the Furstenberg entropy of any (G,µ)-stationary ergodic space is either 0 or larger than ɛ. Virtually free groups, such as SL₂(ℤ), do not have property (T), and neither do their extensions, such as surface groups. For virtually free groups, we construct stationary actions with arbitrarily small, positive entropy. The construction involves building and lifting spaces of lamplighter groups. For some classical lamplighter gropus, these spaces realize a dense set of entropies between 0 and the Poisson boundary entropy.