Subrelations of ergodic equivalence relations

Feldman, Jacob J.; Sutherland, Colin E.; Zimmer, Robert J.

doi:10.1017/s0143385700004958

Cited by 59 publications

(89 citation statements)

References 21 publications

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“…In this subsection, we introduce the notion of normal subgroupoids of a discrete measured groupoid, based on [9] and Subsection 4.6.1 in [27]. This notion is a generalization of normal subrelations of a discrete measured equivalence relation and also a generalization of normal subgroups of a discrete group.…”

Section: Normal Subgroupoidsmentioning

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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Section: Normal Subgroupoidsmentioning

confidence: 99%

Measure equivalence rigidity of the mapping class group

Kida

2010

Ann. Math.

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“…Clearly, the subrelations of finite index are also quasinormal, since the index cocycle as well as every cocycle with values in a finite group is regular. In both cases Q = Q, F = F and Theorem 0.3 gives [FSZ,Theorems 3.1,3.2].…”

Section: Theorem 03 (Classification Of Quasinormal Subrelations)mentioning

confidence: 98%

“…Equivalently, there are choice functions {φ j } j∈J with φ j ∈ N [S], j ∈ J. If, in addition, R is hyperfinite, then by [FSZ,§2] there is a countable amenable group Q ⊂ N [S] with Q ∩ [S] = 1 Q and such that R is generated by S and Q. Definition 2.1.…”

Section: Quasinormal Subrelationsmentioning

confidence: 99%

“…By Proposition 1.8 there exists an index cocycle σ : R → Σ(J) such that the subrelation P := {(x, y) ∈ R | σ(x, y) = Id} is ergodic. Replacing, if necessary, S by a R-conjugated subrelation we may assume that σ is determined by a family of choice functions {φ j } j∈J with the properties: σ(x, φ j (x))(0) = j for all x ∈ X, j ∈ J and S = {(x, y) ∈ R | σ(x, y)(0) = 0} (see [FSZ,Theorem 1.6]). Clearly, P ⊂ S and φ j ∈ N [P], j ∈ J.…”

Section: From Now On R Is An Ergodic Hyperfinite Equivalence Relationmentioning

confidence: 99%

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Quasinormal subrelations of ergodic equivalence relations

Danilenko

1998

Proc. Amer. Math. Soc.

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Abstract. We introduce a notion of quasinormality for a nested pair S ⊂ R of ergodic discrete hyperfinite equivalence relations of type II 1 . (This is a natural extension of the normality concept due to Feldman-Sutherland-Zimmer.) Such pairs are characterized by an irreducible pair F ⊂ Q of countable amenable groups or rather (some special) their Polish closure F ⊂ Q. We show that "most" of the ergodic subrelations of R are quasinormal and classify them. An example of a nonquasinormal subrelation is given. We prove as an auxiliary statement that two cocycles of R with dense ranges in a Polish group are weakly equivalent. IntroductionIt is well known that two ergodic finite measure-preserving actions of countable amenable groups are orbit equivalent [Dy], [CFW]. This can be rephrased in equivalent terms of measured equivalence relations [FM]: there exists a unique (up to isomorphism) hyperfinite discrete ergodic equivalence relation, say R, of type II 1 . A natural subsequent problem that arises here is to study subrelations of R and this is the main concern of the present paper.It was shown in [FSZ] how to associate a countable index set J and a cocycle σ : R → Σ(J) to any pair S ⊂ R of discrete ergodic type II 1 equivalence relations, where Σ(J) is the full permutation group of J. The cardinality of J is called the index of S ⊂ R and is related closely to the Jones index in the study of subvon-Neumann-algebras [Jo]. The cocycle σ is called the index cocycle of S ⊂ R. The weak equivalence class of σ depends only on the isomorphism class of the pair S ⊂ R.J. Feldman, C. E. Sutherland and R. J. Zimmer provided an elegant classification of ergodic hyperfinite pairs S ⊂ R in the following two cases: (a) S is normal, (b) S is of finite index in R [FSZ]. Remark that the case (b) was considered earlier by M. Gerber in a different context-she classified the finite extensions of ergodic probability-preserving transformations up to the "factor orbit equivalence" [Ge]. The purpose of this paper is to extend the above results to a wider class of subrelations, namely quasinormal ones.We call S quasinormal if σ (or its restriction to S) is regular, i.e. σ is cohomologous to a cocycle with dense range in a closed subgroup of Σ(J). The concept of

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“…First we establish the basic terminology and notation concerning Borel equivalence relations that will be needed later (cf. [10,11,47]). …”

Section: Introductionmentioning

confidence: 99%

Symmetric Gibbs measures

Petersen

Schmidt

1997

Trans. Amer. Math. Soc.

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Abstract. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, 'canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

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Subrelations of ergodic equivalence relations

Cited by 59 publications

References 21 publications

Measure equivalence rigidity of the mapping class group

Measure equivalence rigidity of the mapping class group

Quasinormal subrelations of ergodic equivalence relations

Symmetric Gibbs measures

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