A Convex Structure on Sofic Embeddings

TL;DR

This paper extends a convex structure to sofic group representations and characterizes extreme points via ergodic actions of the commutant on the Loeb measure space.

Contribution

It introduces a convex structure on the set of sofic representations and links extremality to ergodic actions of the commutant.

Findings

Extreme points correspond to representations with ergodic commutant actions.

Provides a convex framework for analyzing sofic group representations.

Connects ergodic theory with the structure of sofic embeddings.

Abstract

Nathanial Brown introduced a convex-like structure on the set of unitary equivalence classes of unital *-homomorphisms of a separable type II_1 factor into R^\omega (ultrapower of the hyperfinite factor). The goal of this paper is to introduce such a structure on the set of sofic representations of groups. We prove that if the commutant of a representation acts ergodicaly on the Loeb measure space then that representation is an extreme point.