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arXiv · 2610.29831

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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Laura Ciobanu, Robert D. Gray, Alex Levine. 2026-10-21. The Diophantine Problem with constraints in some free-by-cyclic groups. https://arxiv.org/abs/2610.29831

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