Linear Algebraic Groups and Countable Borel Equivalence Relations Scot Adams and Alexander S. Kechris

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:...

Some Applications of the Adams-kechris Technique

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every Σ1 equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence...

Rigidity theorems for actions of product groups and countable Borel equivalence relations

Means on Equivalence Relations

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.

How Many Turing Degrees are There ?

A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence rel...