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The Diophantine Problem in Wreath Products of Groups

  • Michael Catli
  • Alexei Myasnikov
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Abstract

We study the Diophantine problem in restricted wreath products of groups. Write G=Bwr H, where B is the passive factor and H is the active factor. The main results demonstrate that in a wide class of wreath products G, the Diophantine problem is undecidable. First, we show that if both factors are indicable, then G has a retract isomorphic to ℤwrℤ and therefore has undecidable Diophantine problem. Then, we show that for any groups B and H, the Diophantine problems in B and H are polynomial time reducible to the Diophantine problem in G. Finally, we combine e-definability with retractions in order to prove that the Diophantine problem in G is undecidable whenever B is an infinite finitely generated abelian group and H is a finitely generated group containing an infinite e-definable finitely generated abelian subgroup A.

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Cite

BibTeXRIS
BibTeX
@misc{catli2026diophantine,
  title = {{The Diophantine Problem in Wreath Products of Groups}},
  author = {Michael Catli and Alexei Myasnikov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22719},
  eprint = {2610.22719},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22719},
}
RIS
TY  - UNPB
TI  - The Diophantine Problem in Wreath Products of Groups
AU  - Michael Catli
AU  - Alexei Myasnikov
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22719
UR  - https://arxiv.org/abs/2610.22719
AB  - We study the Diophantine problem in restricted wreath products of groups. Write G=Bwr H, where B is the passive factor and H is the active factor. The main results demonstrate that in a wide class of wreath products G, the Diophantine problem is undecidable. First, we show that if both factors are indicable, then G has a retract isomorphic to ℤwrℤ and therefore has undecidable Diophantine problem. Then, we show that for any groups B and H, the Diophantine problems in B and H are polynomial time reducible to the Diophantine problem in G. Finally, we combine e-definability with retractions in order to prove that the Diophantine problem in G is undecidable whenever B is an infinite finitely generated abelian group and H is a finitely generated group containing an infinite e-definable finitely generated abelian subgroup A.
ER  -