# Countable Borel equivalence relations

Bonn Logic Seminar, Bonn, May 2009

*Abstract*: Borel equivalence relations is an area of descriptive set theory which concerns the complexity of equivalence relations on a standard Borel space (i.e., a Polish space equipped just with its $\sigma$-algebra of Borel sets). There are interesting examples from within logic, such as the Turing equivalence relation on $\mathcal P(\omega)$. Moreover, many classification problems for other areas of mathematics can be regarded as equivalence relations on standard Borel spaces. For instance, the classification problem for torsion-free abelian groups of rank n corresponds to the isomorphism equivalence relation on a suitable subspace of $\mathcal P(\mathbb Q^n)$. Both of these examples are instances of countable Borel equivalence relations, that is, equivalence relations that are Borel as subsets of the plane and which have the property that every equivalence class is countable. After giving the definitions, I’ll discuss what structure theory exists for these objects. I’ll pay special attention to the example of torsion-free abelian groups, where there are several key applications.</p>