Vaught's conjecture and the Glimm-Effros property for Polish transformation groups

Authors:
Greg Hjorth and Slawomir Solecki

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2623-2641

MSC (1991):
Primary 04A15

Published electronically:
March 10, 1999

MathSciNet review:
1467467

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the original Glimm-Effros 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.*

**[Be]**H. Becker,*Vaught's conjecture for complete left invariant Polish groups*, handwritten notes, University of South Carolina, 1996.**[Ben]**Miroslav Benda,*Remarks on countable models*, Fund. Math.**81**(1973/74), no. 2, 107–119. Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, II. MR**0371634****[BeKe1]**Howard Becker and Alexander S. Kechris,*Borel actions of Polish groups*, Bull. Amer. Math. Soc. (N.S.)**28**(1993), no. 2, 334–341. MR**1185149**, 10.1090/S0273-0979-1993-00383-5**[BeKe2]**Howard Becker and Alexander S. Kechris,*The descriptive set theory of Polish group actions*, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR**1425877****[ChKe]**C. C. Chang and H. J. Keisler,*Model theory*, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Studies in Logic and the Foundations of Mathematics, Vol. 73. MR**0409165****[Ef]**Edward G. Effros,*Polish transformation groups and classification problems*, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp. 217–227. MR**619045****[Ga]**S. Gao, Automorphism groups of countable structures, Journal of Symbolic Logic, vol. 63 (1998), pp. 2623-2641.**[Gl]**James Glimm,*Locally compact transformation groups*, Trans. Amer. Math. Soc.**101**(1961), 124–138. MR**0136681**, 10.1090/S0002-9947-1961-0136681-X**[GrMoRy]**A. Grzegorczyk, A. Mostowski, and C. Ryll-Nardzewski,*Definability of sets in models of axiomatic theories*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**9**(1961), 163–167. MR**0163839****[Ha]**Leo Harrington,*Analytic determinacy and 0^{♯}*, J. Symbolic Logic**43**(1978), no. 4, 685–693. MR**518675**, 10.2307/2273508**[HaKeLo]**L. A. Harrington, A. S. Kechris, and A. Louveau,*A Glimm-Effros dichotomy for Borel equivalence relations*, J. Amer. Math. Soc.**3**(1990), no. 4, 903–928. MR**1057041**, 10.1090/S0894-0347-1990-1057041-5**[HaSa]**L. Harrington and R. Sami,*Equivalence relations, projective and beyond*, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 247–264. MR**567673****[Hj]**G. Hjorth,*Orbit cardinals*, preprint, UCLA, 1996.**[HjKe]**Greg Hjorth and Alexander S. Kechris,*Analytic equivalence relations and Ulm-type classifications*, J. Symbolic Logic**60**(1995), no. 4, 1273–1300. MR**1367210**, 10.2307/2275888**[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. 2623-2641. CMP**98:13****[Ke1]**Alessandro Panconesi and Aravind Srinivasan,*The local nature of Δ-coloring and its algorithmic applications*, Combinatorica**15**(1995), no. 2, 255–280. MR**1337357**, 10.1007/BF01200759**[Ke2]**A. S. Kechris,*Lectures on definable group actions and equivalence relations*, unpublished manuscript, Los Angeles, 1994.**[Mi]**Arnold W. Miller,*On the Borel classification of the isomorphism class of a countable model*, Notre Dame J. Formal Logic**24**(1983), no. 1, 22–34. MR**675914****[MiD]**Douglas E. Miller,*On the measurability of orbits in Borel actions*, Proc. Amer. Math. Soc.**63**(1977), no. 1, 165–170. MR**0440519**, 10.1090/S0002-9939-1977-0440519-8**[Mo]**Yiannis N. Moschovakis,*Descriptive set theory*, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR**561709****[Sa]**Ramez L. Sami,*Polish group actions and the Vaught conjecture*, Trans. Amer. Math. Soc.**341**(1994), no. 1, 335–353. MR**1022169**, 10.1090/S0002-9947-1994-1022169-2**[Si]**Jack H. Silver,*Counting the number of equivalence classes of Borel and coanalytic equivalence relations*, Ann. Math. Logic**18**(1980), no. 1, 1–28. MR**568914**, 10.1016/0003-4843(80)90002-9**[So]**Sławomir Solecki,*Equivalence relations induced by actions of Polish groups*, Trans. Amer. Math. Soc.**347**(1995), no. 12, 4765–4777. MR**1311918**, 10.1090/S0002-9947-1995-1311918-2**[Va]**Robert Vaught,*Invariant sets in topology and logic*, Fund. Math.**82**(1974/75), 269–294. Collection of articles dedicated to Andrzej Mostowski on his sixtieth birthday, VII. MR**0363912**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
04A15

Retrieve articles in all journals with MSC (1991): 04A15

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 90095-1555

Email:
greg@math.ucla.edu

**Slawomir Solecki**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
ssolecki@indiana.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02141-8

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