Torsion-free abelian groups are Borel complete


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.


Gianluca Paolini

University of Torino, Via Carlo Alberto 10, 10123, Italy

Saharon Shelah

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel and Department of Mathematics, Rutgers University, Piscataway, NJ, USA