Undecidability of the Diophantine problem for one-relator groups and one-relation monoids

Abstract

We prove that the Diophantine problem, i.e. the problem of solving equations with coefficients, in one-relator groups and one-relation monoids is undecidable in general. Specifically, we show that for all $d \geq 2$, the Diophantine problem for the Baumslag-Gersten group $\mathcal{B}_d = \operatorname{BG}(1,d)$ and for a closely related one-relation monoid $\mathcal{M}_d$ are both undecidable. Both results are obtained by an encoding of a recent undecidability result due to Rybalov.