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Amenable versus hyperfinite Borel equivalence relations

  • Alexander S. Kechris (a1)

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
  • URL: /core/journals/journal-of-symbolic-logic
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