Kazhdan's Property T and the Geometry of the Collection of Invariant Measures

Abstract

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 $ \Omega = \{ 0,1 \}^G $ , will be denoted by $ (\Omega, 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 if the simplex M G $ (\Omega) $ is the Poulsen simplex (i.e. the ergodic measures are dense in M G $ (\Omega) $ ).¶For G a locally compact, second countable group, we introduce an appropriate G-space $ (\Sigma, G) $ analogous to the G-space $ (\Omega, G) $ and then prove similar results for this more general case.