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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.
ON TENSOR PRODUCTS OF GROUP C∗ALGEBRAS AND RELATED TOPICS
"... Abstract. We discuss properties and examples of discrete groups in connection with their operator algebras and related tensor products. Contents ..."
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Cited by 5 (0 self)
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Abstract. We discuss properties and examples of discrete groups in connection with their operator algebras and related tensor products. Contents
Stable actions of central extensions and relative property (T). arXiv preprint arXiv:1309.3739
, 2013
"... Abstract. Let us say that a discrete countable group is stable if it has an ergodic, free, probabilitymeasurepreserving and stable action. Let G be a discrete countable group with a central subgroup C. We present a sufficient condition and a necessary condition for G to be stable. We show that if ..."
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Abstract. Let us say that a discrete countable group is stable if it has an ergodic, free, probabilitymeasurepreserving and stable action. Let G be a discrete countable group with a central subgroup C. We present a sufficient condition and a necessary condition for G to be stable. We show that if the pair (G,C) does not have property (T), then G is stable. We also show that if the pair (G,C) has property (T) and G is stable, then the quotient group G/C is stable. 1.
INVARIANT MEANS AND THE STRUCTURE OF INNER AMENABLE GROUPS
"... Abstract. We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the f ..."
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Abstract. We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first `2Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first `2Betti number and triviality of cost. Moreover, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G/H of G, beyond which the group H is necessarily weakly normal in G. Even more can be said in the particular case of an atomless mean for the conjugation action – that is, when G is inner amenable. We show that inner amenable groups have cost 1 and moreover they have fixed price. We establish Ufincocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field. We also completely characterize linear groups which are stable in the sense of Jones and Schmidt. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H(R), recently studied by Monod, are stable. This includes nonamenable groups constructed by Monod and by Lodha and Moore, as well as Thompson’s group F.
LIMITS OF BAUMSLAGSOLITAR GROUPS AND DIMENSION ESTIMATES IN THE SPACE OF MARKED GROUPS
, 2010
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