Linear algebraic groups and countable Borel equivalence relations

Adams, Scot; Kechris, Alexander

doi:10.1090/S0894-0347-00-00341-6

Linear algebraic groups and countable Borel equivalence relations
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by Scot Adams and Alexander S. Kechris

J. Amer. Math. Soc. 13 (2000), 909-943

DOI: https://doi.org/10.1090/S0894-0347-00-00341-6

Published electronically: June 23, 2000

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Abstract:

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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