Hostname: page-component-76c49bb84f-887v8 Total loading time: 0 Render date: 2025-07-09T22:49:19.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Rotation groups virtually embed into right-angled rotation groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 June 2025

Anthony Genevois
Anthony Genevois*
Affiliation:
Institut Mathématiques Alexander Grothendieck, University of Montpellier, Place Eugène Bataillon, 34090 Montpellier (France)
*
e-mail: anthony.genevois@umontpellier.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In this article, we generalize this observation to rotation groups, which can be thought of as a common generalization of Coxeter groups and graph products of groups. More precisely, we prove that rotation groups (aka periagroups) virtually embed into right-angled rotation groups (aka graph products of groups).

Keywords

Graph productsquasi-median graphsmediangle graphs

MSC classification

Primary: 20F65: Geometric group theory 20F55: Reflection and Coxeter groups

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 25
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Antolín, Y. and Dreesen, D., The Haagerup property is stable under graph products. Preprint, 2013. arXiv:1305.6748.Google Scholar
Alonso, J., Finiteness conditions on groups and quasi-isometries . J. Pure Appl. Algebra 95(1994), no. 2, 121–129.Google Scholar
Alonso, J., Dehn functions and finiteness properties of graph products . J. Pure Appl. Algebra 107(1996), no. 1, 9–17.Google Scholar
Basarab, Ş., Partially commutative Artin-Coxeter groups and their arboreal structure . J. Pure Appl. Algebra 176(2002), no. 1, 1–25.Google Scholar
Berlai, F. and de la Nuez, J., Linearity of graph products . Proc. Lond. Math. Soc. (3) 124(2022), no. 5, 587–600.Google Scholar
Bandelt, H.-J., Mulder, H., and Wilkeit, E., Quasi-median graphs and algebras . J. Graph Theory 18(1994), no. 7, 681–703.Google Scholar
Cooper, D., Long, D., and Reid, A., Infinite Coxeter groups are virtually indicable . Proc. Edinburgh Math. Soc. (2) 41(1998), no. 2, 303–313.Google Scholar
Cohen, D., Projective resolutions for graph products . Proc. Edinburgh Math. Soc. (2) 38(1995), no. 1, 185–188.Google Scholar
Davis, M. and Januszkiewicz, T., Right-angled Artin groups are commensurable with right-angled Coxeter groups . J. Pure Appl. Algebra 153(2000), no. 3, 229–235.Google Scholar
Dyer, M., Reflection subgroups of Coxeter systems . J. Algebra 135(1990), no. 1, 57–73.Google Scholar
Genevois, A., Cubical-like geometry of quasi-median graphs and applications to geometric group theory. Ph.D. thesis, Université Aix-Marseille, 2017.Google Scholar
Genevois, A., Rotation groups, mediangle graphs, and periagroups: a unified point of view on Coxeter groups and graph products of groups. Preprint, 2022. arXiv:2212.06421.Google Scholar
Genevois, A., Special cube complexes revisited: A quasi-median generalization . Can. J. Math. 75(2023), no. 3, 743–777.Google Scholar
Green, E., Graph products of groups. Ph.D. thesis, University of Leeds, 1990.Google Scholar
Haglund, F. and Wise, D., Coxeter groups are virtually special . Adv. Math. 224(2010), no. 5, 1890–1903.Google Scholar
Reckwerdt, E., Weak amenability is stable under graph products . J. Lond. Math. Soc. (2) 96(2017), no. 1, 133–155.Google Scholar
Soergel, M., A generalization of the Davis-Moussong complex for Dyer groups . J. Comb. Algebra 8(2024), nos. 1–2, 209–249.Google Scholar