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

TL;DR

This paper investigates the entropy of algebraic actions of sofic groups, demonstrating conditions under which topological and measure-theoretic entropies coincide, using a reformulation of doubly-quenched convergence.

Contribution

It introduces a new approach to compare entropies of algebraic actions of sofic groups without explicit entropy calculations, unifying known cases.

Findings

Topological and measure-theoretic entropies agree in many cases.

Method recovers all known examples.

Proofs are direct and avoid explicit entropy computations.

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.