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93
Orbit equivalence rigidity
 Annals of Mathematics
, 1999
"... Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X,µ), and let RΓ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation RΓ on X determ ..."
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Cited by 62 (3 self)
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Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X,µ), and let RΓ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation RΓ on X determines the group Γ and the action (X,µ,Γ) uniquely, up to finite groups. The natural action of SLn(Z) on the ntorus R n /Z n, for n> 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II1, which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.
Processes on unimodular random networks
 In preparation
, 2005
"... Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amen ..."
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Cited by 49 (4 self)
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Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasitransitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
Countable Borel Equivalence Relations
 J. MATH. LOGIC
"... This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related s ..."
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Cited by 45 (7 self)
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This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This study is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras
An uncountable family of non orbit equivalent actions of Fn
 J. Amer. Math. Soc
, 2005
"... Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i =1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi,µi) areorbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic, more precisely, if there exists a m ..."
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Cited by 40 (12 self)
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Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i =1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi,µi) areorbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic, more precisely, if there exists a measurable, almost everywhere defined isomorphism f: X1 → X2 such that f∗µ1 = µ2 and the Γ1orbit of µ1almost every x ∈ X1 is sent by f onto the Γ2orbit of f(x). The theory of orbit equivalence, although underlying the “group measure space construction ” of Murray and von Neumann [MvN36], was born with the work of H. Dye who proved, for example, the following striking result [Dy59]: Any two ergodic p.m.p. free actions of Γ1 � Z and Γ2 � � j∈N Z/2Z are orbit equivalent. Through a series of works, the class of groups Γ2 satisfying Dye’s theorem gradually increased until it achieved the necessary and sufficient condition: Γ2 is infinite amenable [OW80]. In particular, all infinite amenable groups produce one and only one ergodic p.m.p. free action up to orbit equivalence (see also [CFW81] for a more
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 40 (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.
Cocycle and orbit equivalence superrigidity for malleable actions of wrigid groups
"... Abstract. We prove that if a countable discrete group Γ is wrigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a f ..."
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Cited by 34 (7 self)
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Abstract. We prove that if a countable discrete group Γ is wrigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any Vvalued measurable cocycle for a measure preserving action Γ � X of Γ on a probability space (X, µ) which is weak mixing on H and smalleable (e.g. the Bernoulli action Γ � [0,1] Γ) is cohomologous to a group morphism of Γ into V. We use the case V discrete of this result to prove that if in addition Γ has no nontrivial finite normal subgroups then any orbit equivalence between Γ � X and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. There has recently been increasing interest in the study of measure preserving actions of groups on (nonatomic) probability spaces up to orbit equivalence (OE), i.e. up to isomorphisms of probability spaces taking the orbits of one action onto the orbits of
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 27 (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
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 26 (6 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