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53
On the superrigidity of malleable actions with spectral gap
 J. Amer. Math. Soc
"... Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certa ..."
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Cited by 79 (7 self)
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Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L ∞ X ⋊Γ ≃ L ∞ Y ⋊Λ comes from a conjugacy of the actions. 1.
Deformation and rigidity for group actions and von Neumann algebras
, 2007
"... We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors wit ..."
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Cited by 64 (7 self)
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We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.
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 53 (16 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
Property (T) and rigidity for actions on Banach spaces
 BHV] [BoS] [Bou] [BuSc] [BuSc’] [BrSo] [C] M. B. Bekka, P. de la Harpe, Alain Valette. “Kazhdan’s
, 2005
"... Abstract. We study property (T) and the fixed point property for actions on L p and other Banach spaces. We show that property (T) holds when L 2 is replaced by L p (and even a subspace/quotient of L p), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for L ..."
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Cited by 52 (6 self)
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Abstract. We study property (T) and the fixed point property for actions on L p and other Banach spaces. We show that property (T) holds when L 2 is replaced by L p (and even a subspace/quotient of L p), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for L p follows from property (T) when 1 < p < 2 +ε. For simple Lie groups and their lattices, we prove that the fixed point property for L p holds for any 1 < p < ∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.
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
ORBIT INEQUIVALENT ACTIONS OF NONAMENABLE GROUPS
, 2008
"... Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such ..."
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Cited by 28 (3 self)
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Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such that E d ∆ ⊂ E c Γ and d has a as a factor. This generalizes the standard notion of coinduction of actions of groups from actions of subgroups. We then use this construction to show that if Γ is a countable nonamenable group, then Γ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.
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.