Linear algebraic groups and countable Borel equivalence relations

Abstract

If R i R_i is an equivalence relation on a standard Borel space B i ( i = 1 , 2 ) B_i\ (i=1,2) , then we say that R 1 R_1 is Borel reducible to R 2 R_2 if there is a Borel function f : B 1 → B 2 f: B_1\to B_2 such that ( x , y ) ∈ R 1 ⇔ ( f ( x ) , f ( y ) ) ∈ R 2 (x,y)\in R_1 \Leftrightarrow (f(x),f(y))\in R_2 . An equivalence relation R R on a standard Borel space B B is Borel if its graph is a Borel subset of B × B 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 Σ 2 1 {\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.