KAZHDANS 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 metriza ble 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 Omega = {0, 1}(G), will be denoted by (Omega, G). We prove the follow ing results. (i) If G has property T then for every (topological) G-ac tion (X, G), M-G(X), when non-empty, is a Bauer simplex (i.e, the set of ergodic measures (extreme points) in MG(X) is closed). (ii) G does not have property T iff the simplex M-G(Omega) is the Poulsen simplex (i.e. the ergodic measures are dense in M-G(Omega)). For G a locally c ompact, second countable group, we introduce an appropriate G-space (S igma, G) analogous to the G-space (Omega, G) and then prove similar re sults for this more general case.