Journal of the American Mathematical Society

Author(s): Scot Adams; Alexander S. Kechris
Journal: J. Amer. Math. Soc. 13 (2000), 909-943.
MSC (2000): Primary 03E15; Secondary 37A20
Posted: June 23, 2000
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is an equivalence relation on a standard Borel space

, then we say that

is Borel reducible to

if there is a Borel function $f: B_1\to B_2$ such that $(x,y)\in R_1 \Leftrightarrow (f(x),f(y))\in R_2$ . An equivalence relation

on a standard Borel space

is Borel if its graph is a Borel subset of $B\times B$ . It is countable if each of its equivalence classes is countable. We investigate the complexity of Borel reducibility of countable Borel equivalence relations on standard Borel spaces. We show that it is at least as complex as the relation of inclusion on the collection of Borel subsets of the real line. We also show that Borel reducibility is ${\boldsymbol\Sigma}^{\boldsymbol 1}_{\boldsymbol 2}$ -complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.

1.: S. Adams, Indecomposability of treed equivalence relations, Israel J. Math., 86 (3), 362-380, 1988. MR 90m:28022
2.: S. Adams, Trees and amenable equivalence relations, Erg. Th. and Dyn. Syst., 10, 1-14, 1990. MR 91d:28041
3.: P. de la Harpe and A. Valette, La Propriété (T) de Kazhdan pour les Groupes Localement Compacts, Astérisque, 175, Soc. Math. de France, 1989. MR 90m:22001
4.: J. Dieudonné, Sur les Groupes Classiques, Hermann, Paris, 1973. MR 49:9094
5.: R. Dougherty, S. Jackson, and A.S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc., 341 (1), 193-225, 1994. MR 94c:03066
6.: E.G. Effros, Transformation groups and C-algebras, Ann. of Math., 81, 38-55, 1965. MR 30:5175
7.: J. Feldman, P. Hahn, and C.C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. in Math., 26, 186-230, 1979. MR 58:1121
8.: J. Feldman and C.C. Moore, Ergodic equivalence relations and von Neumann algebras, I, Trans. Amer. Math. Soc., 234, 289-324, 1977. MR 58:28261a
9.: H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, J. Symb. Logic, 54 (3), 894-914, 1989. MR 91f:03062
10.: L. Fuchs, Infinite Abelian Groups, Academic Press, 1970. MR 41:333
11.: S. Gao, Some applications of the Adams-Kechris technique, preprint, 2000.
12.: J. Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc., 101, 124-138, 1961. MR 25:146
13.: V.V. Gorbatsevich, A.L. Onishchik, and E.B. Vinberg, Foundations of Lie Theory and Lie Transformation Groups, Springer, 1997. MR 99c:22009
14.: L. Harrington, A.S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc., 3 (4), 903-928, 1990. MR 91h:28023
15.: G. Hjorth, Around nonclassifiability for countable torsion-free abelian groups, preprint, 1998.
16.: G. Hjorth and A.S. Kechris, Borel equivalence relations and classifications of countable models, Ann. Pure and Appl. Logic, 82, 221-272, 1996. MR 99m:03073
17.: J.E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math., 21, Springer-Verlag, 1975. MR 53:633
18.: S. Jackson, A.S. Kechris, and A. Louveau, Countable Borel equivalence relations, preprint, 2000.
19.: A.S. Kechris, The structure of Borel equivalence relations in Polish spaces, Set Theory of the Continuum, H. Judah, W. Just, and W.H. Woodin (eds.), MSRI Publications, Springer-Verlag, 26, 89-102, 1992. MR 94h:03093
20.: A.S. Kechris, Countable sections for locally compact group actions, Erg. Th. and Dyn. Syst., 12, 283-295, 1992. MR 94b:22003
21.: A.S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer-Verlag, 1995. MR 96e:03057
22.: A.S. Kechris, Actions of Polish groups and classification problems, preprint, 1998.
23.: A.S. Kechris, New directions in descriptive set theory, Bull. Symb. Logic, 5 (2), 161-174, 1999.
24.: A.S. Kechris, Descriptive dynamics, Descriptive Set Theory and Dynamical Systems, Ed. by M. Foreman, A.S. Kechris, A. Louveau, and B. Weiss, London Math. Society Lecture Note Series, 277, 231-258, Cambridge Univ. Press, 2000.
25.: A. Louveau and B. Velickovic, A note on Borel equivalence relations, Proc. Amer. Math. Soc., 120, 255-259, 1994. MR 94f:54076
26.: G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Erg. der Math. und Ihrer. Grenz., 17, Springer-Verlag, 1991. MR 92h:22021
27.: C.C. Moore, Ergodic theory and von Neumann algebras, Proc. Symp. Pure Math., Amer. Math. Soc., 38, 179-226, 1982. MR 84f:22013
28.: Y.N. Moschovakis, Descriptive Set Theory, North Holland, 1980. MR 82e:03002
29.: J.J. Rotman, An Introduction to the Theory of Groups, Grad. Texts in Math., 148, Springer-Verlag, 1995. MR 95m:20001
30.: K. Schmidt, Algebraic Ideas in Ergodic Theory, CBMS Regional Conference Series in Math., 76, Amer. Math Soc., 1990. MR 92k:28029
31.: J.-P. Serre, A Course in Arithmetic, Grad. Texts. in Math., 7, Springer-Verlag, 1973. MR 49:8956
32.: J. Silver, Counting the number of equivalence classes of Borel and co-analytic equivalence relations, Ann. Math. Logic, 18, 1-28, 1980. MR 81d:03051
33.: S. Thomas, Notes on TFA groups, preprint, 1998.
34.: V.S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc., 109, 191-220, 1963. MR 28:3139
35.: R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, 1984. MR 86j:22014
36.: R. Zimmer, Groups generating transversals to semisimple Lie group actions, Israel J. Math., 73 (2), 151-159, 1991. MR 93e:22012

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 03E15, 37A20

Scot Adams
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: adams@math.umn.edu

Alexander S. Kechris
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
Email: kechris@caltech.edu

DOI: 10.1090/S0894-0347-00-00341-6
PII: S 0894-0347(00)00341-6
Keywords: Borel equivalence relations, Borel reducibility, cocycles, superrigidity
Received by editor(s): March 27, 1999
Received by editor(s) in revised form: April 21, 2000
Posted: June 23, 2000
Additional Notes: The first author's research was partially supported by NSF Grant DMS 9703480.
The second author's research was partially supported by NSF Grant DMS 9619880 and a Visiting Miller Research Professorship at U.C. Berkeley.
Copyright of article: Copyright 2000, American Mathematical Society

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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