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Lookup NU author(s): Dr Andrew Duncan
This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
© 2023 The AuthorsWe generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to right-angled Artin groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a right-angled Artin group G, certain Magnus subgroups embed in the quotient G=G/N(r); that if r=sn has root s in G then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. We also give conditions under which the question of which Magnus subgroups of G embed in G reduces to the same question in the minimal parabolic subgroup of G containing r. In many cases this allows us to characterise Magnus subgroups which embed in G, via a condition on r and the commutation graph of G, and to find further examples of quotients G where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord Γ, almost all cyclically reduced elements of the right-angled Artin group G(Γ) satisfy the conditions of the theorem.
Author(s): Duncan AJ, Juhasz A
Publication type: Article
Publication status: Published
Journal: Journal of Algebra
Year: 2023
Volume: 622
Pages: 506-555
Print publication date: 15/05/2023
Online publication date: 17/02/2023
Acceptance date: 02/04/2022
Date deposited: 29/03/2023
ISSN (print): 0021-8693
ISSN (electronic): 1090-266X
Publisher: Academic Press Inc.
URL: https://doi.org/10.1016/j.jalgebra.2023.02.012
DOI: 10.1016/j.jalgebra.2023.02.012
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