Sofic entropy and amenable groups

TL;DR

This paper proves that for amenable groups, the recently introduced sofic entropy coincides with classical entropy, using new invariants and existing theorems to establish the equivalence.

Contribution

It demonstrates that sofic entropy equals classical entropy for amenable group actions, bridging a gap between these concepts with new invariants and methods.

Findings

Sofic entropy equals classical entropy for amenable groups

Introduction of upper-sofic entropy as a measure-conjugacy invariant

Application of Rudolph and Weiss's theorem to establish the result

Abstract

In previous work, I introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit change $\sigma$-algebra.