Classificazione dei Gruppi Abeliani Privi di Torsione a meno di Isomorfismo

This thesis aims to study the article ¿The classification problem for torsion-free abelian groups of finite rank¿ of Simon Thomas which analyzes the complexity of the isomorphism relation on the collection of torsion-free abelian groups of finite rank, showing why no satisfactory system of complete invariants has yet been found for higher rank than 1. Using Descriptive Set Theory and Zimmer's Superrigidity Theory in the purely Borel setting, our ultimate goal is to show that the complexity of the isomorphism relation for torsion-free abelian groups of rank n increases strictly with the rank n.