Sofic mean dimension

TL;DR

This paper introduces a new concept called sofic mean dimension for group actions, extending previous notions to sofic groups, and demonstrates its usefulness in differentiating actions with infinite entropy.

Contribution

It generalizes mean dimension to sofic groups, providing a new tool for analyzing and distinguishing complex group actions beyond amenable groups.

Findings

Defines sofic mean dimension for group actions.

Shows its relation to existing mean dimensions for amenable groups.

Demonstrates its effectiveness in distinguishing actions with infinite entropy.

Abstract

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.