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PreprintCC BY 4.0arXiv

The Diophantine Problem with constraints in some free-by-cyclic groups

  • Laura Ciobanu
  • Robert D. Gray
  • Alex Levine
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Abstract

In this paper we explore the Diophantine Problem, that is, the solvability of equations, in one-relator and free-by-cyclic groups, showing that this is decidable in some cases and undecidable otherwise, depending on their abelianisation and centre. In particular, we focus on a version of the Diophantine Problem which asks about the existence of solutions satisfying abelianisation constraints, and prove a number of results concerning the decidability of this problem for free-by-cyclic groups with centre, tree-GBS groups, cyclically pinched groups and the first hydra group.

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Cite

BibTeXRIS
BibTeX
@misc{ciobanu2026diophantine,
  title = {{The Diophantine Problem with constraints in some free-by-cyclic groups}},
  author = {Laura Ciobanu and Robert D. Gray and Alex Levine},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29831},
  eprint = {2610.29831},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29831},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Diophantine Problem with constraints in some free-by-cyclic groups
AU  - Laura Ciobanu
AU  - Robert D. Gray
AU  - Alex Levine
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29831
UR  - https://arxiv.org/abs/2610.29831
AB  - In this paper we explore the Diophantine Problem, that is, the solvability of equations, in one-relator and free-by-cyclic groups, showing that this is decidable in some cases and undecidable otherwise, depending on their abelianisation and centre. In particular, we focus on a version of the Diophantine Problem which asks about the existence of solutions satisfying abelianisation constraints, and prove a number of results concerning the decidability of this problem for free-by-cyclic groups with centre, tree-GBS groups, cyclically pinched groups and the first hydra group.
ER  -