Measure Conjugacy Invariants for Actions of Countable Sofic Groups

This paper is motivated by an old and central problem in measurable dynamics: given two dynamical systems, determine whether or not they are measurably conjugate, i.e., isomorphic. Let us set some notation. A dynamical system (or system for short) is a triple (G,X, μ), where (X,μ) is a probability space and G is a group acting by measure-preserving transformations on (X,μ). We will also call this a dynamical system over G, a G-system or an action of G. In this paper, G will always be a discrete countable group. Two systems (G,X, μ) and (G, Y, ν) are isomorphic (i.e., measurably conjugate) if and only if there exist conull sets X ′ ⊂ X,Y ′ ⊂ Y and a bijective measurable map φ : X ′ → Y ′ such that φ−1 : Y ′ → X ′ is measurable, φ∗μ = ν and φ(gx) = gφ(x)∀g ∈ G, x ∈ X ′. A special class of dynamical systems called Bernoulli systems or Bernoulli shifts has played a significant role in the development of the theory as a whole because it was the problem of trying to classify them that motivated Kolmogorov to introduce the mean entropy of a dynamical system over Z [Ko58, Ko59]. That is, Kolmogorov defined for every system (Z, X, μ) a number h(Z, X, μ) called themean entropy of (Z, X, μ) that quantifies, in some sense, how “random” the system is. His definition was modified by Sinai [Si59]; the latter has become standard. Bernoulli shifts also play an important role in this paper, so let us define them. Let (K,κ) be a standard Borel probability space. For a discrete countable group G, let K = ∏ g∈G K be the set of all functions x : G → K with the product Borel structure and let κ be the product measure on K. The group G acts on K by (gx)(f) = x(g−1f) for x ∈ K and g, f ∈ G. This action is measure-preserving. The system (G,K, κ) is the Bernoulli shift over G with base (K,κ). It is nontrivial if κ is not supported on a single point. Before Kolmogorov’s seminal work [Ko58, Ko59], it was unknown whether all nontrivial Bernoulli shifts over Z were measurably conjugate to each other. He proved that h(Z,K, κ) = H(κ) where H(κ), the entropy of κ, is defined as follows. If there exists a finite or countably infinite setK ′ ⊂ K such that κ(K ′) = 1,