Linear algebraic groups and countable Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). For this to be of any interest both I and c must be explicit or definable and as simple and concrete as possible. The theory of Borel equivalence relations studies the set-theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems.