Membership problems in braid groups and Artin groups

Gray, Robert D. and Nyberg-Brodda, Carl-Fredrik (2025) Membership problems in braid groups and Artin groups. Journal of the London Mathematical Society, 112 (4). ISSN 0024-6107

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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 Bn if and only if n ≤ 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.

Item Type:	Article
Additional Information:	Funding information: The first author is supported by the EPSRC Fellowship grant EP/V032003/1 “Algorithmic, topological and geometric aspects of infinite groups, monoids and inverse semigroups.” The second author is supported by the Mid-Career Researcher Program (RS-2023-00278510) through the National Research Foundation funded by the government of Korea, and by the KIAS Individual Grant MG094701 at Korea Institute for Advanced Study.
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups:	Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR)
Depositing User:	LivePure Connector
Date Deposited:	29 Sep 2025 11:30
Last Modified:	21 Oct 2025 12:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/100506
DOI:	10.1112/jlms.70326

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