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On the Borel completeness of torsion-free abelian groups and a Slaman-Wehner degree spectrum

  • George Crittenden
  • Matthew Harrison-Trainor
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Abstract

It was recently shown by Paolini and Shelah, with another proof by Laskowski and Ulrich, that torsion-free abelian groups are Borel complete. This means that isomorphism for torsion-free abelian groups is as complicated as possible, so that torsion-free abelian groups do not admit any meaningful complete isomorphism invariants. We give a proof following the ideas of the Paolini-Shelah argument but with several expository advantages. In particular, the combinatorial part of the argument is much simplified. We make use of the ideas of this proof to show that there is a torsion-free abelian group with no computable copy but with a copy computable from every non-computable set. This answers a long-standing question of Downey and Goncharov.

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Cite

BibTeXRIS
BibTeX
@misc{crittenden2026borel,
  title = {{On the Borel completeness of torsion-free abelian groups and a Slaman-Wehner degree spectrum}},
  author = {George Crittenden and Matthew Harrison-Trainor},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05451},
  eprint = {2610.05451},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05451},
}
RIS
TY  - UNPB
TI  - On the Borel completeness of torsion-free abelian groups and a Slaman-Wehner degree spectrum
AU  - George Crittenden
AU  - Matthew Harrison-Trainor
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05451
UR  - https://arxiv.org/abs/2610.05451
AB  - It was recently shown by Paolini and Shelah, with another proof by Laskowski and Ulrich, that torsion-free abelian groups are Borel complete. This means that isomorphism for torsion-free abelian groups is as complicated as possible, so that torsion-free abelian groups do not admit any meaningful complete isomorphism invariants. We give a proof following the ideas of the Paolini-Shelah argument but with several expository advantages. In particular, the combinatorial part of the argument is much simplified. We make use of the ideas of this proof to show that there is a torsion-free abelian group with no computable copy but with a copy computable from every non-computable set. This answers a long-standing question of Downey and Goncharov.
ER  -