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$G\curvearrowright X$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$G\curvearrowright X$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.