Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

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 $\boldsymbol{\Sigma}^1_1$ 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.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03E15

Retrieve articles in all journals with MSC (1991): 03E15


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

American Mathematical Society