  We prove that the Borel space of torsion-free Abelian groups with domain
$\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space
is as complicated as possible, as an isomorphism relation. This solves a
long-standing open problem in descriptive set theory, which dates back to the
seminal paper on Borel reducibility of Friedman and Stanley from 1989.
