Kazhdan's property t and the geometry of the collection of invariant measures

For a countable group G and an action (X, G) of G on a compact metrizable space X, let M_G(X) denote the simplex of probability measures on X invariant under G. The natural action of G on the space of functions Ω = {0, 1}^G, will be denoted by (Ω, G). We prove the following results. (i) If G has property T then for every (topological) G-action (X, G), M_G(X), when non-empty, is a Bauer simplex (i.e. the set of ergodic measures (extreme points) in M_G(X) is closed). (ii) G does not have property T iff the simplex M_G(Ω) is the Poulsen simplex (i.e. the ergodic measures are dense in M_G(Ω)). For G a locally compact, second countable group, we introduce an appropriate G-space (∑, G) analogous to the G-space (Ω, G) and then prove similar results for this more general case.