Torsion-Free Abelian Groups are Borel Complete Article Swipe

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Mathematics Abelian group Borel equivalence relation Borel set Isomorphism (crystallography) Borel hierarchy Torsion (gastropod) Pure mathematics Elementary abelian group Polish space Rank of an abelian group Borel measure Discrete mathematics Mathematical analysis Probability measure Chemistry Crystal structure Surgery Separable space Crystallography Medicine

Gianluca Paolini , Saharon Shelah ·

YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2102.12371 · OA: W4287323722

We prove 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.