Abstract : I will talk about my result joint with S. Shelah establishing that the Borel space of torsion-free Abelian groups with domain ω 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. After this I will survey some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups, and on the problem of classification of countable co-Hopfian abelian and 2-nilpotent groups.

Keywords : Abelian groups; Borel completeness; co-Hopfian groups