On orbit equivalence of measure preserving actions

D. Gaboriau

Article publié :

Rigidity in dynamics and geometry (Cambridge, 2000), pages 167-186, Springer, Berlin, 2002

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We give a brief survey of some classification results on orbit equivalence of probability measure preserving countable group actions. The notion of l2 Betti numbers for groups is gently introduced. An account of orbit equivalence invariance for l2 Betti numbers is presented together with a description of the theory of equivalence relation actions on simplicial complexes. We relate orbit equivalence to a measure theoretic analogue of commensurability and quasi-isometry of groups : measure equivalence. Rather than a complete description of these subjects, a lot of examples are provided.