The submonoid and rational subset membership problems for Artin groups

TL;DR

This paper explores the decidability of submonoid and rational subset membership problems in Artin groups, linking their complexity to the embedding of specific right-angled Artin groups and subgroup separability.

Contribution

It establishes the equivalence of these problems in Artin groups and characterizes their decidability based on the group's embedding properties and subgroup separability.

Findings

Problems are undecidable if the group embeds certain right-angled Artin groups.

Problems are decidable if and only if the group is subgroup separable.

Characterization depends solely on the defining graph of the Artin group.

Abstract

We demonstrate that the submonoid membership problem and the rational subset membership problem are equivalent in Artin groups. Both these problem are undecidable in a given Artin group if and only if the group embeds the right-angled Artin groups of rank 4 over a path or a square; and this can be characterized using only the defining graph of the Artin group. These results generalize the ones by Lohrey - Steinberg for right-angled Artin groups. Moreover, both these decision problems are decidable for a given Artin group if and only if the group is subgroup separable. This equivalence for right-angled Artin groups is provided by Lohrey - Steinberg and Metaftsis - Raptis. The equivalence for general Artin groups comes from some observations here and the characterization of separable Artin groups by Almeida - Lima.