On Krein-Milman theorem for the space of sofic representations

TL;DR

This paper explores the convex structure of the space of sofic representations of groups, demonstrating the existence of many extreme points and complex face structures, including a representation possibly outside the convex hull of extreme points.

Contribution

It establishes that minimal faces in the space are extreme points, constructs numerous extreme points for free groups, and presents a unique representation with unusual convex properties.

Findings

Minimal faces are extreme points in Sof(G)

Uncountably many extreme points for Sof(𝔽₂)

Existence of a representation outside the convex hull of extreme points

Abstract

Denote by $Sof(G)$ the space of sofic representations of a countable group $G$. This space is known by a result of the second author, to have a convex-like structure. We show that, in this space, minimal faces are extreme points. We then construct uncountable many extreme points for $Sof(\mathbb{F}_2)$ and show that there exists a decreasing chain of closed faces with empty intersection. Finally we construct a strangely looking sofic representation in $Sof(\mathbb{F}_2)$ that we believe it is outside of the closure of the convex hull of extreme points.