Abstract
AbstractWe prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.
Highlights
We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, the actions have the same sofic topological entropy
Entropy turns out to be sensitive in meaningful ways to the various kinds of restrictions that one may naturally impose on an orbit equivalence, its role as an invariant remaining intact in some cases but completely neutralized in others. The history of this relationship traces back several decades and in its original thrust encompasses the work of Vershik on actions of locally finite groups [42, 43], the Ornstein isomorphism machinery for Bernoulli shifts [32], the theory of Kakutani equivalence [19, 11, 33, 9], and Kammeyer and Rudolph’s general theory of restricted orbit equivalence for p.m.p. actions of countable amenable groups that all of this inspired [37, 17, 18]
The basic geometric idea at play in Austin’s work when the group is not virtually cyclic is the possibility of finding, within suitable connected Følner subsets of the group, a connected subgraph which is sparse but at the same time dense at a specified coarse scale. By recasting this sparse connectivity as a condition on the action that we called property SC and circumventing the “derandomization” of [3] with its reliance on the Rudolph–Weiss technique, we established in [25] the following extension beyond the amenable setting: if G is a countable group containing a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, H is a countable group, and G (X, μ) and H (Y, ν) are free p.m.p. actions which are Shannon orbit equivalent, the maximum sofic measure entropies of the actions satisfy hν (H Y ) ≥ hμ(G X)
Summary
Throughout the paper G and H are countable discrete groups, with identity elements eG and eH. A continuous action G X on a compact metrizable space is said to be topologically free if the Gδ set of all x ∈ X such that sx = x for all s ∈ G \ {eG} is dense. It is uniquely ergodic if there is a unique G-invariant Borel probability measure on X. By the Riesz representation theorem this is equivalent to the existence of a unique G-invariant state (i.e., unital positive linear functional) for the induced action of G on the C∗-algebra C(X) of continuous functions on X given by (gf )(x) = f (g−1x) for all g ∈ G, f ∈ C(X), and x ∈ X. By Gelfand theory, this is equivalent to the unique ergodicity, in the topological-dynamical sense above, of the induced action of G on the spectrum of L∞(X, μ)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
