Undecidable Diophantine problems in generalisations of one-relator groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917545109356544 |
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| author | Gray, Robert D. Levine, Alex |
| author_facet | Gray, Robert D. Levine, Alex |
| contents | Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag--Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group $G$, with a single fixed finite rank free subgroup $H$, such that the Diophantine problem for $G$ with $H$-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form $F_3 \rtimes F_2$ with an undecidable Diophantine problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Undecidable Diophantine problems in generalisations of one-relator groups Gray, Robert D. Levine, Alex Group Theory 20F05, 20F65, 20F10 Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag--Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group $G$, with a single fixed finite rank free subgroup $H$, such that the Diophantine problem for $G$ with $H$-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form $F_3 \rtimes F_2$ with an undecidable Diophantine problem. |
| title | Undecidable Diophantine problems in generalisations of one-relator groups |
| topic | Group Theory 20F05, 20F65, 20F10 |
| url | https://arxiv.org/abs/2605.30535 |