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Nazarov: Perfect matchings as IID factors on nonamenable groups
 Europ. J. Combin
, 2011
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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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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.
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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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.
WEAK EQUIVALENCE AND NONCLASSIFIABILITY OF MEASURE PRESERVING ACTIONS
"... Abstract. AbértWeiss have shown that the Bernoulli shift sΓ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that sΓ × a is weakly equivalent to a. Using random Ber ..."
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Abstract. AbértWeiss have shown that the Bernoulli shift sΓ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that sΓ × a is weakly equivalent to a. Using random Bernoulli shifts introduced by AbértGlasnerVirag we generalized this to nonfree actions, replacing sΓ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence on the weak equivalence class of a free measure preserving action do not admit classification by countable structures. This in particular shows that there are no free weakly rigid actions, i.e., actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group Γ is said to have property EMD ∗ if the action pΓ of Γ on its profinite completion weakly contains all ergodic measure preserving actions of Γ, and Γ is said to have property MD if ι×pΓ weakly contains all measure preserving actions of Γ, where ι denotes the identity action on a standard nonatomic probability space. Kechris shows that EMD ∗ implies MD and asks if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.
Mixing actions of countable groups are almost free
, 1208
"... Abstract. A measure preserving action of a countably infinite group Γ is called totally ergodic if every infinite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a finit ..."
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Abstract. A measure preserving action of a countably infinite group Γ is called totally ergodic if every infinite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a finite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely on the FeitThompson theorem. As a consequence we deduce a group theoretic characterization of countable groups whose nontrivial Bernoulli factors are all free: these are precisely the groups that posses no finite normal subgroup other than the trivial subgroup. 1.
MODULAR ACTIONS AND AMENABLE REPRESENTATIONS
"... Abstract. Consider a measurepreserving action Γ � (X, µ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ � X ..."
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Abstract. Consider a measurepreserving action Γ � (X, µ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ � X is nonamenable, then there does not exist a countabletoone Borel homomorphism from the orbit equivalence relation of the skew product action Γ � α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countablysplitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, nonamenable, residually finite group induces at least three mutually orbit inequivalent free, measurepreserving, ergodic actions as well as two nonBorel bireducible ones. 1.
A Spectral Strong Approximation Theorem for Measure Preserving Actions
, 2014
"... Let be a finitely generated group acting by probability measure preserving maps on the standard Borel space (X;). We show that if H is a subgroup with relative spectral radius greater than the global spectral radius of the action, then H acts with finitely many ergodic components and spectral ga ..."
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Let be a finitely generated group acting by probability measure preserving maps on the standard Borel space (X;). We show that if H is a subgroup with relative spectral radius greater than the global spectral radius of the action, then H acts with finitely many ergodic components and spectral gap on (X;). This answers a question of Shalom who proved this for normal subgroups.
EXAMPLES OF MIXING SUBALGEBRAS OF VON NEUMANN ALGEBRAS AND THEIR NORMALIZERS
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