# Weak containment and maximal sofic approximations

## TL;DR

This paper explores properties of sofic actions, demonstrating their closure under direct products, the existence of a maximal element, and establishing entropy and Pinsker factor formulas for specific sofic approximations.

## Contribution

It introduces a class of sofic actions with a maximal element and constructs special sofic approximations that unify measure and topological entropy.

## Key findings

- Sofic actions are closed under direct products.
- Existence of a maximal sofic action in the weak containment order.
- Equality of sofic measure entropy and topological entropy for certain algebraic actions.

## Abstract

We show that the class of sofic actions is closed under direct products and contains a (non-unique) maximal element in the weak containment order. For any sofic group we construct nice sofic approximations such that all the sofic actions are approximable by them in a doubly-quenched way. We use recent result by Hayes to establish for these sofic approximations the equality of sofic measure entropy to the topological one for algebraic actions whenever the former is not $-\infty$. We also use his another result to establish the product formula for Pinsker factors of these approximations.