A word-hyperbolic special monoid with undecidable Diophantine problem

TL;DR

This paper presents a specific word-hyperbolic, special monoid with a trivial group of units where the Diophantine problem is undecidable, demonstrating limits of previous decidability results in hyperbolic groups.

Contribution

It provides the first example of a word-hyperbolic monoid with undecidable Diophantine problem, answering open questions and highlighting differences from hyperbolic groups.

Findings

Decidable in hyperbolic groups but undecidable in certain monoids

Constructs a simple, finitely presented, special monoid with undecidable problem

Answers open questions from Garreta & Gray (2019)

Abstract

The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special monoid $M$ with trivial group of units, and such that the Diophantine problem is undecidable in $M$. This answers two questions asked by Garreta & Gray in 2019, and shows that the decidability of the Diophantine problem in hyperbolic groups, as proved by Dahmani & Guirardel, does not generalise to word-hyperbolic monoids.