Membership problems in braid groups and Artin groups

TL;DR

This paper classifies decision problems in braid and Artin groups, identifying which are decidable based on the structure of the groups, and resolves an open problem for braid groups with up to three strands.

Contribution

It provides a complete classification of decidability for several decision problems in Artin groups and braid groups, extending previous classifications and solving an open problem.

Findings

Decidable submonoid membership in Artin groups characterized by forbidden subgraphs.

Submonoid membership in braid groups is decidable if and only if n ≤ 3.

Generalization of previous results on right-angled Artin groups.

Abstract

We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the non-existence of certain forbidden induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, and the fixed-target submonoid membership problem. In the case of braid groups our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group $\mathbf{B}_n$ if and only if $n \leq 3$, which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey & Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.