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Linear algebraic groups and countable Borel equivalence relations |
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| Authors: | Scot Adams Alexander S Kechris |
| Institution: | Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 ; Department of Mathematics, Caltech, Pasadena, California 91125 |
| Abstract: | If |
| Keywords: | Borel equivalence relations Borel reducibility cocycles superrigidity |
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is an equivalence relation on a standard Borel space 
, then we say that 
is Borel reducible to 
if there is a Borel function 
such that 
. An equivalence relation 
on a standard Borel space 
is Borel if its graph is a Borel subset of 
. 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 
-complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.