On Krein-Milman theorem for the space of sofic representations

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.