A Choquet simplex is a compact convex set in which every point is the barycenter of a unique boundary measure. Infinite-dimensional simplices are quite diverse, e.g., any Polish space can be obtained as the set of extreme points of a metrizable Choquet simplex. However, those that arise naturally in functional analysis and ergodic theory tend to be either Bauer (i.e., the extreme points form a compact set) or the Poulsen simplex (the unique metrizable simplex whose extreme points are dense).
A famous result of Glasner and Weiss captures a precise instance of this dichotomy: for any countable group G, the simplex Pr_G(2^G) of invariant probability measures of the topological Bernoulli shift is either Bauer or the Poulsen simplex. Moreover, it is Bauer if and only if G has Property (T).
I will discuss a generalization of the Glasner--Weiss Theorem to a larger class of simplices arising from permutation groups. This addresses questions of Austin motivated by the theory of exchangeable random variables. The proof is based on continuous model theory, and more precisely on recent developments in affine logic. | 
