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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Kechris, Alexander S. and Miller, Benjamin D. 2008. Means on equivalence relations. Israel Journal of Mathematics, Vol. 163, Issue. 1, p. 241.


    JACKSON, S. KECHRIS, A. S. and LOUVEAU, A. 2002. COUNTABLE BOREL EQUIVALENCE RELATIONS. Journal of Mathematical Logic, Vol. 02, Issue. 01, p. 1.


    Thomas, Simon 2002. On the complexity of the classification problem for torsion-free abelian groups of rank two. Acta Mathematica, Vol. 189, Issue. 2, p. 287.


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Amenable versus hyperfinite Borel equivalence relations

  • Alexander S. Kechris (a1)
  • DOI: http://dx.doi.org/10.2307/2275102
  • Published online: 01 March 2014
Abstract

Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [x]E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.

The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is called hyperfinite if it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relation E induced by a Borel action of a countable amenable group G on X and for every (Borel) probability measure μ on X, there is a Borel invariant set YX with μ(Y) = 1 such that EY (= the restriction of E to Y) is hyperfinite. (Recall that a countable group G is amenable if it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether every E induced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true for G = ℤn. However, the problem is still open even for abelian G. Our main purpose here is to provide a weaker affirmative answer for general amenable G (and more—see below). We need a definition first. Given two standard Borel spaces X, Y, a universally measurable isomorphism between X and Y is a bijection ƒ: XY such that both ƒ, ƒ-1 are universally measurable. (As usual, a map g : ZW, with Z and W standard Borel spaces, is called universally measurable if it is μ-measurable for every probability measure μ on Z.) Notice now that to assert that a countable Borel equivalence relation on X is hyperfinite is trivially equivalent to saying that there is a standard Borel space Y and a hyperfinite Borel equivalence relation F on Y, which is Borel isomorphic to E, i.e., there is a Borel bijection ƒ: XY with xEy ⇔ ƒ(x)F ƒ(y). We have the following theorem.

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[AL]S. Adams and R. Lyons , Amenability, Kazhdan's property, and percolation for trees, groups and equivalence relations, Israel Journal of Mathematics, vol. 75 (1991), pp. 341370.

[CFW]A. Connes , J. Feldman , and B. Weiss , An amenable equivalence relation is generated by a single transformation, Ergodic Theory and Dynamical Systems, vol. 1 (1981), pp. 430450.

[FM]J. Feldman and C. C. Moore , Ergodic equivalence relations, cohomology and von Neumann algebras, I., Transactions of the American Mathematical Society, vol. 234 (1977), pp. 289324.

[HKL]L. Harrington , A. S. Kechris , and A. Louveau , A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), pp. 903927.

[K2]A. S. Kechris , The structure of Borel equivalence relations in Polish spaces, Set Theory and the Continuum (H. Judah , W. Just , and W. H. Woodin , editors), Mathematical Sciences Research Institute Publications, Springer, New York, 1992, pp. 89102.

[OW]D. Ornstein and B. Weiss , Ergodic theory of amenable group actions I: The Rohlin lemma, Bulletin of the American Mathematical Society, vol. 2 (1980), pp. 161164.

[SS]T. Slaman and J. Steel , Definable functions on degrees, Cabal Seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer-Verlag, Berlin and New York, 1988, pp. 3755.

[V]V. S. Varadarajan , Groups of automorphisms of Borel spaces, Transactions of the American Mathematical Society, vol. 109 (1963), pp. 191220.

[W2]B. Weiss , Measurable dynamics, Conference in modern analysis and probability (R. Beals et al., editors), Contemporary Mathematics, vol. 26, American Mathematical Society, Providence, Rhode Island, 1984, pp. 395421.

[Z]R. Zimmer , Hyperfinite factors and amenable ergodic actions, Inventiones Mathematicae, vol. 41 (1977), pp. 2331.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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