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Analytic equivalence relations and Ulm-type classifications

Published online by Cambridge University Press:  12 March 2014

Greg Hjorth  and
Alexander S. Kechris
Greg Hjorth
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90024, E-mail: greg@cco.caltech.edu
Alexander S. Kechris
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125, E-mail: kechris@math.caltech.edu
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Extract

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.

The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:

(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else

(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).

Moreover, alternative (II) is equivalent to:

(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 4 , December 1995 , pp. 1273 - 1300
Copyright
Copyright © Association for Symbolic Logic 1995

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