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41
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 91 (4 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.
Rank gradient, cost of groups and the rank versus Heegard genus problem
, 2008
"... We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a byproduct, we show that the ‘Rank vs. Heegaard genus’ conjecture on hyperbolic 3manifolds is incompatible with the ‘Fixed Price problem’ in topological dynamics. ..."
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Cited by 31 (2 self)
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We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a byproduct, we show that the ‘Rank vs. Heegaard genus’ conjecture on hyperbolic 3manifolds is incompatible with the ‘Fixed Price problem’ in topological dynamics.
On the Complexity of the Isomorphism Relation for Finitely Generated Groups
, 1998
"... Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two genera ..."
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Cited by 24 (11 self)
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Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators. The proofs are grouptheoretic, and we refer to descriptive set theory only for the relevant definitions and for motivation for the results. Introduction Given a class K of structures for a fixed first order language L, one may ask what kinds of complete invariants can be used to classify the elements of K up to isomorphism. For those classes consisting of the countable models of some L ! 1 ;! sentence, Friedman and Stanley [FS] proposed to use the methods of descriptive set theory to study their possible invariants and defined the notion of Borel reducibility between such classes of structures. In [HK], Hjorth and Kechris continued this study and situated it within the general the...
Outer automorphism groups of some ergodic equivalence relations
 Comment. Math. Helv
, 2005
"... Abstract. Let R a be countable ergodic equivalence relation of type II1 on a standard probability space (X, µ). The group Out R of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map Rclasses to Rclasses modulo those which preserve almost every ..."
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Cited by 22 (2 self)
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Abstract. Let R a be countable ergodic equivalence relation of type II1 on a standard probability space (X, µ). The group Out R of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map Rclasses to Rclasses modulo those which preserve almost every Rclass. We analyze the group Out R for relations R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of Out R and explicitly computing Out R for the standard actions. The method is based on Zimmer’s superrigidity for measurable cocycles, Ratner’s theorem and Gromov’s Measure Equivalence construction. 1. Introduction and Statement
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.
On Popa’s cocycle superrigidity theorem
 Int. Math. Res. Not. IMRN, (19):Art. ID rnm073
"... Abstract. These notes contain an Ergodictheoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of “rigid ” grou ..."
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Cited by 14 (0 self)
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Abstract. These notes contain an Ergodictheoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of “rigid ” groups) satisfy the assumptions of Popa’s theorem. 1. Introduction and Statement
On affine actions of Lie groups
, 1998
"... this paper, we obtain, by elementary geometrical and "algebraic" methods some new results and new proofs of existing results and (partial) answers to some (partial) questions. ..."
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Cited by 8 (1 self)
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this paper, we obtain, by elementary geometrical and "algebraic" methods some new results and new proofs of existing results and (partial) answers to some (partial) questions.