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 that G acts on by automorphisms. Using (and reformulating) the notion of doubly-quenched convergence developed by Austin, we show that in many cases the topological and the measure-theoretic entropy with respect to the Haar measure of this action 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.