Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups

Abstract

Let G be a sofic group and X a compact group with G X by automorphisms. Using (and reformulating) the notion of local and doubly empirical convergence developed by Austin, we show that in many cases the topological and the measure-theoretic entropy with respect to the Haar measure of G X agree. Our method of proof recovers all known examples. Moreover, the proofs are direct and do not go through explicitly computing the measure-theoretic or topological entropy.

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

“…This paper furthers the study of the entropy theory on algebraic actions of sofic groups along the lines we previously developed in [30,31]. Given a countable, discrete, group G, an algebraic action of G is an action G X by continuous automorphisms of a compact, metrizable group X.…”

“…This general condition is that of strong soficity (see the discussion before Corollary 2.23 for the definition of strong soficity), and under this condition we have the following theorem which is a combination of work in [30,31].…”

“…Theorem 1.1 (Theorem 1.1 in [30], Corollary 1.4 in [31]). Suppose G is a countable, discrete, sofic group with sofic approximation σ k : G → S d k .…”

“…is strongly sofic if and only if there is a sequence of measures µ k ∈ Prob(X d k ) which are asymptotically supported on topological microstates and locally weak * -converge to m X (by [7,Corollary 5.7] and [30,Corollary 2.14]). Thus we aim to show that existence of a sequence of measures µ k ∈ Prob(X d k ) which are asymptotically supported on topological microstates and locally weak * -converge to m X .…”

A Brief Introduction to Sofic Entropy Theory

“…There were computations of entropy showing that it was true in a number of special cases [KL11b,Bow11a,BL12,Hay16b] but these all proceeded by computing the topological entropy and the measure entropy separately in terms of analytic data and then showing their equality. So it is astonishing that just recently Ben Hayes proved under a mild hypothesis on the actions that the topological entropy agrees with the measure entropy [Hay16a]. The proof uses Austin's lde-sofic-entropy [Aus16a].…”