Torsion-free abelian groups are Borel complete

We prove that the Borel space of torsion-free Abelian groups with domain 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.