Borel superrigidity and the classification problem for the torsion-free abelian groups of finite rank

In 1937, Baer solved the classification problem for the torsion-free abelian groups of rank 1. Since then, despite the efforts of many mathematicians, no satisfactory solution has been found of the classification problem for the torsion-free abelian groups of rank n ≥ 2. So it is natural to ask whether the classification problem for the higher rank groups is genuinely difficult. In this article, I will explain how this question can be partially answered, using ideas from descriptive set theory and Zimmer’s superrigidity theory. Mathematics Subject Classification (2000). Primary 03E15, 20K15, 37A20.