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Torsion-free Abelian groups are Borel complete

By Gianluca Paolini

Appears in collection : XVI International Luminy Workshop in Set Theory / XVI Atelier international de théorie des ensembles

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.

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Citation data

  • DOI 10.24350/CIRM.V.19809603
  • Cite this video Paolini Gianluca (9/14/21). Torsion-free Abelian groups are Borel complete. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19809603
  • URL https://dx.doi.org/10.24350/CIRM.V.19809603



  • Paolini, Gianluca, and Saharon Shelah. "Torsion-Free Abelian Groups are Borel Complete." arXiv preprint arXiv:2102.12371 (2021). - https://arxiv.org/abs/2102.12371
  • Paolini, Gianluca, and Saharon Shelah. "On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups." arXiv preprint arXiv:2107.11290 (2021). - https://arxiv.org/abs/2107.11290
  • Gianluca Paolini and Saharon Shelah. "Co-Hopfian Groups are Complete co-Analytic". In preparation. -

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