# Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

## TL;DR

This paper demonstrates that certain one-relator inverse monoids and groups have undecidable word and submonoid membership problems, respectively, advancing understanding of computational limits in algebraic structures.

## Contribution

It proves the existence of one-relator inverse monoids with undecidable word problems and shows that some one-relator groups have undecidable submonoid membership problems, answering longstanding open questions.

## Key findings

- Existence of one-relator inverse monoids with undecidable word problem
- Existence of one-relator groups with undecidable submonoid membership problem
- Addresses a problem posed in 1987 by Margolis, Meakin, and Stephen

## Abstract

We prove the following results: (1) There is a one-relator inverse monoid $\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The first of these results answers a problem originally posed by Margolis, Meakin and Stephen in 1987.