Abstract
We work with a conjecture on the BNS-invariant of Artin groups stated by Almeida and Kochloukova. We show that the class of Artin groups that satisfy this conjecture is closed under finite graph products. As a consequence, we show that the conjecture is true for all Artin groups of finite type and other subclasses.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almeida, K.: The BNS-invariant for some Artin groups of arbitrary circuit rank. J. Group Theory 20(4), 793–806 (2017)
Almeida, K.: The BNS-invariant for Artin groups of circuit rank 2. J. Group Theory 21(2), 189–228 (2018)
Almeida, K., Kochloukova, D.H.: The \(\Sigma ^1\)-invariant for some Artin groups of rank 3 presentation. Comm. Algebra 43(2), 702–718 (2015)
Almeida, K., Kochloukova, D.H.: The \(\Sigma ^1\)-invariant for Artin groups of circuit rank 1. Forum Math. 27(5), 2901–2925 (2015)
Bieri, R., Geoghegan, R., Kochloukova, D.: The sigma invariants of Thompson’s group F. Groups Geom. Dyn. 4(2), 263–273 (2010)
Bieri, R., Neumann, W.D., Strebel, R.: A geometric invariant of discrete groups. Invent. Math. 90, 451–477 (1987)
Bieri, R., Renz, B.: Valuations on free resolutions and higher geometric invariants of groups. Comment. Math. Helv. 63, 464–497 (1988)
Kochloukova, D.: On the \(\Sigma ^2\)-invariants of the generalised R. Thompson groups of type \(F\). J. Algebra 371, 430–456 (2012)
Meier, J.: Geometric invariants for Artin groups. Proc. Lond. Math. Soc. 74(1), 151–173 (1997)
Meier, J., VanWyk, L.: The Bieri-Neumann-Strebel invariants for graph groups. Proc. London Math. Soc. 71(2), 263–280 (1995)
Meier, J., Meinert, H., VanWyk, L.: On the \(\Sigma \)-invariants of Artin Groups. Topol. Appl. 110, 71–81 (2001)
Meinert, H.: The Bieri–Neumann–Strebel invariant for graph products of groups. J. Pure Appl. Algebra 103(2), 205–210 (1995)
Mulholland, J., Rolfsen, D.: Local indicability and commutator subgroups of Artin groups. arXiv:math/0606116 (2006)
Zaremsky, M.: On the \(\Sigma \)-invariants of generalized Thompson groups and Houghton groups. Int. Math. Res. Not. 19, 5861–5896 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
de Almeida, K.E., Lima, F.F. Finite graph product closure for a conjecture on the BNS-invariant of Artin groups. Arch. Math. 116, 131–139 (2021). https://doi.org/10.1007/s00013-020-01530-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-020-01530-8