Vaught's conjecture and the GlimmEffros property for Polish transformation groups
Authors:
Greg Hjorth and Slawomir Solecki
Journal:
Trans. Amer. Math. Soc. 351 (1999), 26232641
MSC (1991):
Primary 04A15
Published electronically:
March 10, 1999
MathSciNet review:
1467467
Fulltext PDF Free Access
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Abstract: We extend the original GlimmEffros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.
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 H. Becker, Vaught's conjecture for complete left invariant Polish groups, handwritten notes, University of South Carolina, 1996.
 [Ben]
 M. Benda, Remarks on countable models, Fundamenta Mathematicae, vol. 81 (1974), pp. 26232641. MR 51:7852
 [BeKe1]
 H. Becker and A. S. Kechris, Borel actions of Polish groups, Bulletin of the American Mathematical Society, vol. 28 (1993), pp. 26232641. MR 93m:03083
 [BeKe2]
 H. Becker and A. S. Kechris, The descriptive set theory of Polish groups actions, Cambridge, London Mathematical Society Lecture Note Series, 1997. MR 98d:54068
 [ChKe]
 C. C. Chang and H. J. Keisler, Model theory, Amsterdam, NorthHolland, 1973. MR 53:12927
 [Ef]
 E. G. Effros, Polish transformation groups and classification problems, General topology and modern analysis, Rao and McAuley (eds.), New York, Academic Press, 1981, pp. 217227. MR 82k:54064
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 S. Gao, Automorphism groups of countable structures, Journal of Symbolic Logic, vol. 63 (1998), pp. 26232641.
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 J. Glimm, Locally compact transformation groups, Transactions of the American Mathematical Society, vol. 101 (1961), pp. 26232641. MR 25:146
 [GrMoRy]
 A. Grzegorczyk, A. Mostowski, C. RyllNardzewski, Definability of sets of models of axiomatic theories, Bulletin of the Polish Academy of Sciences (Mathematics, Astronomy and Physics), vol. 9 (1961), pp. 26232641. MR 29:1138
 [Ha]
 L. Harrington, Analytic determinacy and , Journal of Symbolic Logic, vol. 43 (1978), pp. 26232641. MR 80b:03065
 [HaKeLo]
 L. Harrington, A. S. Kechris, A. Louveau, A GlimmEffros dichotomy theorem for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), pp. 26232641. MR 91h:28023
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 L. Harrington and R. Sami, Equivalence relations, projective and beyond, Logic Colloquium'78, Amsterdam, NorthHolland, 1979, pp. 247264. MR 82d:03080
 [Hj]
 G. Hjorth, Orbit cardinals, preprint, UCLA, 1996.
 [HjKe]
 G. Hjorth and A. S. Kechris, Analytic equivalence relations and Ulmtype classifications, Journal of Symbolic Logic., vol. 60(1995), pp. 26232641. MR 96m:54068
 [HjKeLo]
 G. Hjorth, A.S. Kechris, and A. Louvaeu, Borel equivalence relations induced by actions of the symmetric group, Annals of Pure and Applied Logic, vol. 92 (1998), pp. 26232641. CMP 98:13
 [Ke1]
 A. S. Kechris, Classical descriptive set theory, New York, SpringerVerlag, 1995. MR 96e:05057
 [Ke2]
 A. S. Kechris, Lectures on definable group actions and equivalence relations, unpublished manuscript, Los Angeles, 1994.
 [Mi]
 A. Miller, On the Borel classification of the isomorphism type of a countable model, Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 26232641. MR 84c:03055
 [MiD]
 D. Miller, On the measurability of orbits in Borel actions, Proceedings of the American Mathematical Society, vol. 63(1977), pp. 26232641. MR 55:13394
 [Mo]
 Y. N. Moschovakis, Descriptive set theory, Amsterdam, NorthHolland, 1980. MR 82e:03002
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 R. L. Sami, Polish group actions and the Vaught conjecture, Transactions of the American Mathematical Society, vol. 341 (1994), pp. 335353. MR 94c:03068
 [Si]
 J. H. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Mathematical Logic, vol. 18 (1980), pp. 26232641. MR 81d:03051
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 S. Solecki, Equivalence relations induced by actions of Polish groups, Transactions of the American Mathematical Society, 347 (1995), pp. 26232641. MR 96c:03100
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Additional Information
Greg Hjorth
Affiliation:
Department of Mathematics, 253–37, California Institute of Technology, Pasadena, California 91125
Address at time of publication:
Department of Mathematics, MSB 6363, University of California, Los Angeles, California 900951555
Email:
greg@math.ucla.edu
Slawomir Solecki
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email:
ssolecki@indiana.edu
DOI:
http://dx.doi.org/10.1090/S0002994799021418
PII:
S 00029947(99)021418
Keywords:
Polish group,
orbit equivalence relation
Received by editor(s):
August 18, 1995
Received by editor(s) in revised form:
June 16, 1997
Published electronically:
March 10, 1999
Article copyright:
© Copyright 1999
American Mathematical Society
