On sofic approximations of $\mathbb F_2\times\mathbb F_2$

TL;DR

This paper constructs a novel sofic approximation of the free group product that challenges previous assumptions, answering a question posed by Bowen.

Contribution

It provides the first example of a sofic approximation of _2 _2 that is not derived from a branched cover of homomorphism-based approximations.

Findings

Constructed a non-branched-cover sofic approximation of _2 _2

Answered Bowen's question on the nature of sofic approximations

Demonstrated the diversity of sofic approximations beyond traditional methods

Abstract

We construct a sofic approximation of $\mathbb F_2\times\mathbb F_2$ that is not essentially a "branched cover" of a sofic approximation by homomorphisms. This answers a question of L. Bowen.