Furstenberg entropy realizations for virtually free groups and lamplighter groups

Abstract

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.

Property (T) and the Furstenberg entropy of nonsingular actions

“…A particularly interesting class of nonsingular actions are the stationary ones. Furstenberg entropy has been a useful tool in the study of stationary actions (e.g., [5,8]), and the study of the set of entropy values realizable by properly nonsingular stationary actions has attracted some interest [11,2,6].…”

“…Nevo's theorem (Theorem A.1) implies that for every (G, µ), where G has property (T), the Furstenberg entropy values of properly nonsingular, ergodic µ-stationary actions are bounded away from zero; in this case we say that (G, µ) has a stationary entropy gap. In [2] and [6] it is shown that some (G, µ) without property (T) have no stationary entropy gap. However, this is not a characterization of property (T): in a previous paper [3, Proposition 7.5] we show that there exist (G, µ) without property (T), but with a stationary entropy gap.…”

Unimodularity of invariant random subgroups

“…Although measure preserving G-actions and their stabilizers have been studied for some time (e.g., [9,23]), IRSs were first introduced by Abért, Glasner and Virág [3], and simultaneously by Vershik [26] under a different name. Since then, IRSs have appeared in a number of papers, either as direct subjects of study [12][13][14]18,26], as probabilistic limits of manifolds with increasing volume [1], or as tools to understand stationary group actions [11,17].…”

Generic stationary measures and actions

“…By contrast, if G is non-amenable then an invariant measure need not exist. However, there are surprisingly few explicit constructions of stationary actions: aside from measure-preserving actions and Poisson boundaries, there are constructions from invariant random subgroups [10,31] and methods for combining stationary actions via joinings [22]. There is a general structure theory of stationary actions [22] and a very deep structure theory in the case that G is a higher rank semisimple Lie group [45,46].…”

“…It is known that some non property (T) groups do not have an entropy gap (e.g. free groups [10], some lamplighter groups, and SL 2 (Z) [31]). However, we will next describe groups with an entropy gap which fail to have property (T).…”