Some applications of the Adams-Kechris technique

Author:
Su Gao

Journal:
Proc. Amer. Math. Soc. **130** (2002), 863-874

MSC (1991):
Primary 03E15

DOI:
https://doi.org/10.1090/S0002-9939-01-06082-8

Published electronically:
June 20, 2001

MathSciNet review:
1866043

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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence relations and derive analogous results for the classification problem of Borel automorphisms.

**1.**S. ADAMS AND A. S. KECHRIS,*Linear algebraic groups and countable Borel equivalence relations,*J. Amer. Math. Soc. 13 (2000), 909-943. CMP**2000:16****2.**H. BECKER AND A. S. KECHRIS,*The descriptive set theory of Polish group actions,*London Mathematical Society Lecture Notes Series 232, Cambridge University Press, 1996. MR**98d:54068****3.**J. D. CLEMENS,*Borel automorphisms,*handwritten notes, 1999.**4.**S. EIGEN, A. HAJIAN, AND B. WEISS,*Borel automorphisms with no finite invariant measure,*Proc. Amer. Math. Soc. 126 (1998), no. 12, 3619-3623. MR**99b:28024****5.**I. FARAH,*Ideals induced by Tsirelson submeasures,*Fund. Math. 159 (1999), no. 3, 243-258. CMP**99:11****6.**I. FARAH,*Basis problem for turbulent actions I: Tsirelson submeasures,*Annals of Pure and Applied Logic, to appear.**7.**G. HJORTH,*Classification and orbit equivalence relations,*Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, 2000. MR**2000k:03097****8.**G. HJORTH AND A. S. KECHRIS,*The complexity of the classification of Riemann surfaces and complex manifolds,*Illinois J. Math. 44 (2000), no. 1, 104-137. MR**2000m:03115****9.**A. S. KECHRIS,*Classical descriptive set theory,*Springer-Verlag, New York, 1995. MR**96e:03057****10.**A. LOUVEAU AND B. VELICKOVIC,*A note on Borel equivalence relations,*Proc. Amer. Math. Soc. 120 (1994), no. 1, 255-259. MR**94f:54076****11.**B. VELICKOVIC,*A note on Tsirelson type ideals,*Fund. Math. 159 (1999), no. 3, 259-268. MR**2000f:03142**

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Additional Information

**Su Gao**

Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Email:
sugao@its.caltech.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06082-8

Keywords:
Borel equivalence relations,
Borel (bi-)reducibility

Received by editor(s):
February 10, 2000

Received by editor(s) in revised form:
August 13, 2000, and August 23, 2000

Published electronically:
June 20, 2001

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2001
American Mathematical Society