On the Diophantine problem in some one-relator groups

TL;DR

This paper investigates the decidability of the Diophantine problem in various one-relator groups, using algebraic methods to classify groups where equations can be algorithmically solved, including hyperbolic and torus knot groups.

Contribution

It introduces new classes of groups related to one-relator groups, proves decidability results for specific cases, and connects the problem to right-angled Artin groups and hyperbolic groups.

Findings

Decidability of the Diophantine problem in certain one-relator groups.

Undecidability of the submonoid membership problem in G_{2,2}.

All Newman groups NG(p,q) are hyperbolic and have decidable Diophantine problems.

Abstract

We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister--Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation $a^mb^n = 1$ is virtually a direct product of hyperbolic groups for all $m, n \geq 0$, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the two-generator, one-relator groups $G_{m,n}$ with defining relation a commutator $[a^m, b^n] = 1$, where $m, n \geq 1$. In doing so, we define and study a natural class of groups (RABSAGs), related to right-angled Artin groups (RAAGs). We reduce the Diophantine problem in the groups $G_{m,n}$ to the Diophantine problem in groups which are virtually certain RABSAGs. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group $G_{2,2}$ with the single defining relation $[a^2, b^2] = 1$. We use the recent classification by Gray & Howie of RAAG subgroups of one-relator groups to classify the RAAG subgroups of some RABSAGs, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups $\operatorname{NG}(p,q)$, which are $(p+1)$-generated one-relator groups generalising the solvable Baumslag--Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.