On Groups of Measure Preserving Transformations. I

Dye, Heather A.

doi:10.2307/2372852

Cited by 387 publications

(215 citation statements)

References 3 publications

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“…actions are orbit equivalent if and only if their associated crossed product von Neumann algebras are isomorphic by an isomorphism that preserves the Cartan subalgebras [Si55]. H. Dye [Dy59], [Dy63] proved the pioneering result that any two ergodic p.m.p. actions of the group of integers on the unit interval are OE.…”

Section: Introductionmentioning

confidence: 99%

Stable orbit equivalence of Bernoulli shifts over free groups

Bowen¹

2011

Groups Geom. Dyn.

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Abstract. Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if G 1 , G 2 are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over G 1 is stably orbit equivalent to every nontrivial Bernoulli shift over G 2 . This answers a question of S. Popa.Mathematics Subject Classification (2010). 37A20.

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Section: Introductionmentioning

confidence: 99%

Stable orbit equivalence of Bernoulli shifts over free groups

Bowen¹

2011

Groups Geom. Dyn.

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“…The first magnificent result on OE is due to Ornstein and Weiss [36] following Dye [5], [6] and it can be stated in terms of ME as follows: A discrete group is ME to ‫ޚ‬ if and only if it is an infinite amenable group. Connes, Feldman and Weiss [4] obtain a generalized result in terms of amenable discrete measured equivalence relations.…”

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confidence: 99%

Measure equivalence rigidity of the mapping class group

Kida

2010

Ann. Math.

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We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

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“…actions of any two infinite amenable groups and are orbit equivalent-a result proved by Dye in the case and are Abelian [8] and by Ornstein-Weiss in general ([29], see also [6])) implies that the above question is well-posed only for non-amenable groups. For a non-amenable group , it is known that admits at least two non-OE actions [5,18,36].…”

Section: Introductionmentioning

confidence: 99%

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$

Ioana

2010

Invent. math.

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We prove that if a countable group contains a copy of F 2 , then it admits uncountably many non orbit equivalent actions. IntroductionThroughout this paper we consider free, ergodic, measure preserving (m.p.) actions (X, μ) of countable, discrete groups on standard probability spaces (X, μ). Measurable group theory is roughly the study of such group actions from the viewpoint of the induced orbit equivalence relation. A basic question in measurable group theory is to find groups which admit many non-orbit equivalent actions (see the survey [35]). In this respect, recall that two free, ergodic, m.p. actions (X, μ) and (Y, ν) are said to be orbit equivalent (OE) if they induce isomorphic equivalence relations, i.e. if there exists a measure space isomorphism θ :The striking lack of rigidity manifested by amenable groups (any two free, ergodic m.p. actions of any two infinite amenable groups and are orbit equivalent-a result proved by Dye in the case and are Abelian [8] and by Ornstein-Weiss in general ([29], see also [6])) implies that the above question is well-posed only for non-amenable groups. For a non-amenable group , it is known that admits at least two non-OE actions [5,18,36]. Moreover, recently, the following classes of non-amenable groups have been shown to admit uncountably many non-OE actions: property (T) groups [18], free groups A. Ioana ( )

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On Groups of Measure Preserving Transformations. I

Cited by 387 publications

References 3 publications

Stable orbit equivalence of Bernoulli shifts over free groups

Stable orbit equivalence of Bernoulli shifts over free groups

Measure equivalence rigidity of the mapping class group

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$

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