Results 1  10
of
41
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 ..."
Abstract

Cited by 79 (7 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 68 (9 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 64 (7 self)
 Add to MetaCart
(Show Context)
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.
Examples of groups that are measure equivalent to the free group. Ergodic Theory Dynam
 Systems
, 2005
"... Measure Equivalence (ME) is the measure theoretic counterpart of quasiisometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a n ..."
Abstract

Cited by 28 (2 self)
 Add to MetaCart
(Show Context)
Measure Equivalence (ME) is the measure theoretic counterpart of quasiisometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable freefactor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups.
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 ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
(Show Context)
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
Nazarov: Perfect matchings as IID factors on nonamenable groups
 Europ. J. Combin
, 2011
"... ar ..."
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. ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
(Show Context)
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.