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26
A converse to Dye's theorem
"... Every nonamenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit ..."
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Cited by 49 (2 self)
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Every nonamenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable nonhyperfinite Borel equivalence relation which is not universal for treeable in the ^B ordering.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 41 (7 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
Some computations of 1cohomology groups and construction of non orbit equivalent actions
"... Abstract. For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H × K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measurepreserving actions σΛ of G on the probability space such that the 1’st ..."
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Cited by 35 (10 self)
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Abstract. For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H × K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measurepreserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H 1 (σΛ, G), is equal to Char(G) × Λ. We deduce that G has uncountably many non stably orbit equivalent actions. We also calculate 1cohomology groups and show existence of “many ” non stably orbit equivalent actions for free products of groups as above. Let G be a countable discrete group and σ: G → Aut(X, µ) a free measure preserving (m.p.) action of G on the probability space (X, µ), which we also view as an integral preserving action of G on the abelian von Neumann algebra A = L ∞ (X, µ). A 1cocycle for (σ, G) is a map w: G → U(A), satisfying wgσg(wh) = wgh, ∀g, h ∈ G, where
Ergodic Subequivalence Relations Induced by a Bernoulli Action, available at arXiv: 0802.2353
"... Abstract. Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ � [0, 1] Γ, where [0,1] Γ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation R of S, there exists a partition {Xi} i≥0 of [0, 1] Γ with Rinvariant ..."
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Cited by 25 (4 self)
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Abstract. Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ � [0, 1] Γ, where [0,1] Γ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation R of S, there exists a partition {Xi} i≥0 of [0, 1] Γ with Rinvariant measurable sets such that R X0 is hyperfinite and R Xi is strongly ergodic (hence ergodic), for every i ≥ 1. §1. Introduction and statement of results. During the past decade there have been many interesting new directions arising in the field of measurable group theory. One direction came from the deformation/rigidity theory developed recently by S. Popa in order to study group actions and von Neumann algebras ([P5]). Using this theory, Popa obtained striking rigidity
Orbit Equivalence and Measured Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICIANS (ICM), HYDERABAD: INDIA
, 2010
"... We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions. ..."
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Cited by 20 (0 self)
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We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.
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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Cited by 18 (2 self)
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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.
A SURVEY OF MEASURED GROUP THEORY
"... Abstract. The title refers to the area of research which studies infinite groups using measuretheoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence betw ..."
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Cited by 18 (1 self)
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Abstract. The title refers to the area of research which studies infinite groups using measuretheoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.
A construction of nonorbit equivalent actions of F_n
, 2006
"... Given 2 ≤ n ≤ ∞, we construct a concrete 1parameter family of orbit inequivalent actions of Fn. These actions arise as diagonal products between a generalized Bernoulli action and the action Fn � (T 2, λ 2), where Fn is seen as a subgroup of SL(2, Z). ..."
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Cited by 13 (1 self)
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Given 2 ≤ n ≤ ∞, we construct a concrete 1parameter family of orbit inequivalent actions of Fn. These actions arise as diagonal products between a generalized Bernoulli action and the action Fn � (T 2, λ 2), where Fn is seen as a subgroup of SL(2, Z).
Invariant cocycles, random tilings and the superK and strong Markov properties
, 1996
"... We consider 1cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension defined by the cocycle is ergodic. A ..."
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Cited by 8 (3 self)
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We consider 1cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many of the results in [9] to higherdimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.