On the Krein–Milman theorem for the space of sofic representations

Abstract

Denote by $\mathrm{Sof}(G)$ 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 uncountably many extreme points for $\mathrm{Sof}({\mathbb{F}}_r)$ and show that there exists a decreasing chain of closed faces with empty intersection. Finally, we construct a strange-looking sofic representation in $\mathrm{Sof}({\mathbb{F}}_r)$ that we believe to be outside of the closure of the convex hull of extreme points.