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Linear algebraic groups and countable Borel equivalence relations

Authors: Scot Adams and Alexander S. Kechris
Journal: J. Amer. Math. Soc. 13 (2000), 909-943
MSC (2000): Primary 03E15; Secondary 37A20
Published electronically: June 23, 2000
MathSciNet review: 1775739
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Abstract | References | Similar Articles | Additional Information


If $R_i$ is an equivalence relation on a standard Borel space $B_i (i=1,2)$, then we say that $R_1$ is Borel reducible to $R_2$ 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 $R$ on a standard Borel space $B$ 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.

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Additional Information

Scot Adams
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Alexander S. Kechris
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125

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
Published electronically: 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.
Article copyright: © Copyright 2000 American Mathematical Society