Countable Borel equivalence relations, Borel reducibility, and orbit equivalence

ing from the proof given above for Gaboriau-Popa we obtain theorems such as: Theorem 2.10 Let (X, d) be a complete, separable metric space equipped with an atomless Borel probability measure μ. Suppose Γ acts ergodically by measure preserving transformations on (X,μ) and the action on (X, d) is expansive. Let (Et)0<t<1 be a collection of distinct countable Borel equivalence relations on X with: (a) on any non-null measurable A ⊂ X we have for all s 6= t and a.e. x ∈ A {y ∈ A : yEsx} 6= {y ∈ A : yEtx}; (b) each Es ⊃ EΓ. Then each Es is orbit equivalent to only countably many Et’s. I mention this not as an exercise in proof mining, but rather because it might be interesting to explore dynamical properties such as expansiveness. Conceivably there could be other applications to the theory of countable Borel equivalence relations. For instance, just for example, I do not know which countable groups admit measure preserving expansive actions. 2.6 Countable Borel equivalence relations up to Borel reducibility For a long while we only had finitely many countable Borel equivalence relations which were known to be distinct in the ≤B ordering. This was finally settled by Adams and Kechris: Theorem 2.11 (Adams-Kechris, [1]) There are continuum many ≤B-incomparable countable Borel equivalence relations. Their argument ultimately relied on super rigidity results for certain classes of algebraic groups, as developed in [29]. For the free groups there is no hint that such a rigidity theory should exist, and by implication there is not even the slightest suggestion that there is a parallel theory for treeable equivalence relations.