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.
Fields of Study & Subjects
Advanced Operator Algebra Research
Geometric and Algebraic Topology
Advanced Topology and Set Theory
Publication details
- Journal / source
- Groups Geometry and Dynamics
- Volume
- 20
- Issue
- 2
- Pages
- 381–401
- Publication date
- 12 February 2026
- Language
- EN