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Amenable actions and almost invariant sets
 Proc. Amer. Math. Soc
"... Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In parti ..."
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Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets. 1.
MODULAR ACTIONS AND AMENABLE REPRESENTATIONS
"... Abstract. Consider a measurepreserving action Γ � (X, µ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ � X ..."
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Abstract. Consider a measurepreserving action Γ � (X, µ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ � X is nonamenable, then there does not exist a countabletoone Borel homomorphism from the orbit equivalence relation of the skew product action Γ � α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countablysplitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, nonamenable, residually finite group induces at least three mutually orbit inequivalent free, measurepreserving, ergodic actions as well as two nonBorel bireducible ones. 1.