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arXiv · 2608.01983

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

Carl-Fredrik Nyberg-Brodda
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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.

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Carl-Fredrik Nyberg-Brodda. 2026-08-03. Undecidability of the Diophantine problem for one-relator groups and one-relation monoids. https://arxiv.org/abs/2608.01983

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