The complexity of isomorphism between countably based profinite groups

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based profinite group as a point in a Polish space. Then we study the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs.