Geometric & Functional Analysis GAFA

, Volume 7, Issue 5, pp 917–935 | Cite as

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

  • E. Glasner
  • B. Weiss

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.

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Copyright information

© Birkhäuser Verlag, Basel 1997

Authors and Affiliations

  • E. Glasner
    • 1
  • B. Weiss
    • 2
  1. 1.Eli Glasner, Sackler Faculty of Exact Sci., Tel Aviv University, Tel Aviv, Israel, e-mail: glasner@math.tau.ac.il IL
  2. 2.Benjamin Weiss, Mathematical Institute, Hebrew University, Jerusalem, Israel, e-mail: weiss@math.huji.ac.il IL

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