Krieger’s finite generator theorem for actions of countable groups I

Abstract

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért-Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.h Rok G (X, µ) = inf H(α|I ) : α is a countable partition and σ-alg G (α)∨I = B(X) , where I is the σ-algebra of G-invariant sets. In this paper, we will only be interested in ergodic actions, in which case the Rokhlin entropy simplifies to h Rok G (X, µ) = inf H(α) : α is a countable generating partition .

On a measurable analogue of small topological full groups

“…Seward also has a quantitative version of the above result, yielding a generating observable whose range is as small as possible (see [Sew18], where he also defines Rokhlin entropy).…”

“…The main goal of the present paper is to extend this result to general measure-preserving actions of finitely generated groups. In order to do so, the right notion of entropy is Rokhlin entropy, which was recently introduced by Seward [Sew18]. It is the infimum of the entropies of generating partitions and, for Z-actions, it coincides with the usual definition of entropy of a measure-preserving transformation by a result of Rokhlin [Rok67].…”

An Announce of Results Linking Kolmogorov Complexity to Entropy for Amenable Group Actions

“…Strictly speaking, the theorem above is a combination of corollary 2.7 from [SeTD] which states that the so-called Rokhlin entropy is equal to the Kolmogorov-Sinai entropy for free actions of an amenable group, and the main result of [Se14], asserting that the statement of the theorem above holds after substituting the Kolmogorov-Sinai entropy with the Rokhlin entropy.…”

“…I consider this bound to be the main contribution of this paper. The proof has three main ingredients: a Shannontype frequency bound for asymptotic complexity; the monotonicity of asymptotic complexity under measurable equivariant maps; the recent developements in generating partition theory due to Seward [Se14], Seward and Tucker-Drob [SeTD]. The much weaker "mean" form of the bound was used by Bernshteyn in [Be16].…”

Positive entropy actions of countable groups factor onto Bernoulli shifts

“…In this paper, we will work with a different notion of entropy which was introduced by the author in 2014 [47] and is defined for actions of arbitrary (not necessarily sofic) countable groups. If G is a countable group, G (X, µ) is an ergodic p.m.p.…”