Work

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

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.29831

Authors: Ciobanu, Laura, Gray, Robert D., Levine, Alex

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Deutsche Forschungsgemeinschaft, Engineering and Physical Sciences Research Council

Awards: G1372115314, G2033548659, G6966102507, G7201794056, EXC 2046: MATH+: Berlin Mathematics Research Center, Algorithmic, topological and geometric aspects of infinite groups, monoids and inverse semigroups

OpenAlex ID: W7223224990

Cites: 4

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Related to: 10