Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T13:14:00.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Embedding right-angled Artin groups into Brin–Thompson groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  23 April 2019

JAMES BELK ,
COLLIN BLEAK  and
FRANCESCO MATUCCI
JAMES BELK
Affiliation:
University of St Andrews e-mail: jmb42@st-andrews.ac.uk
COLLIN BLEAK
Affiliation:
University of St Andrews e-mail: jmb42@st-andrews.ac.uk
FRANCESCO MATUCCI
Affiliation:
University of Campinas e-mail: francesco@ime.unicamp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every finitely-generated right-angled Artin group embeds into some Brin–Thompson group nV. It follows that any virtually special group can be embedded into some nV, a class that includes surface groups, all finitely-generated Coxeter groups, and many one-ended hyperbolic groups.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20F36: Braid groups; Artin groups 20E07: Subgroup theorems; subgroup growth
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 225 - 229
Copyright
© Cambridge Philosophical Society 2019

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.)

Footnotes

†

The first and second authors have been partially supported by EPSRC grant EP/R032866/1 during the creation of this paper.

‡

The third author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM) and gratefully acknowledges the support of the Fundação para a Ciência e a Tecnologia (FCT projects UID/Multi/04621/2013 (CEMAT-Ciências) and PEst-OE/MAT/UI0143/2014).

References

REFERENCES

Agol, I.. The virtual Haken conjecture. Doc. Math. 18 (2013), 1045–1087.Google Scholar
Bleak, C. & Salazar–Díaz, O.. Free products in R. Thompson’s group V. Trans. Amer. Math. Soc. 365.11 (2013), 5967–5997.CrossRefGoogle Scholar
Bridson, M.. On the subgroups of right-angled Artin groups and mapping class groups. Math. Res. Lett. 20.2 (2013), 203–212.10.4310/MRL.2013.v20.n2.a1CrossRefGoogle Scholar
Brin, M.. Higher dimensional Thompson groups. Geom. Dedicata 108 (2004), 163–192.10.1007/s10711-004-8122-9CrossRefGoogle Scholar
Charney, R.. An introduction to right-angled Artin groups. Geom. Dedicata 125.1 (2007), 141–158.10.1007/s10711-007-9148-6CrossRefGoogle Scholar
Corwin, N. & Haymaker, K.. The graph structure of graph groups that are subgroups of Thompson’s group V. Internat. J. Algebra Comput. 26.8 (2016), 1497–1501.CrossRefGoogle Scholar
Crisp, J. & Farb, B.. The prevalence of surface groups in mapping class groups. Preprint.Google Scholar
Crisp, J. & Wiest, B.. Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups. Algebr. Geom. Topol. 4.1 (2004), 439–472.CrossRefGoogle Scholar
Haglund, F. & Wise, D.. Special cube complexes. Geom. Funct. Anal. 17.5 (2008), 1551–1620.CrossRefGoogle Scholar
Haglund, F. & Wise, D.. Coxeter groups are virtually special. Adv. Math. 224.5 (2010), 1890–1903.CrossRefGoogle Scholar
Kato, M.. Embeddings of right-angled Artin groups into higher dimensional Thompson groups. J. Algebra Appl. 17.8 (2018), 1850159.CrossRefGoogle Scholar
Koberda, T.. Ping-pong lemmas with applications to geometry and topology. IMS Lecture Notes, Singapore (2012).Google Scholar
Krasner, M. & Kaloujnine, L.. Produit complet des groupes de permutations et problème d’extension de groupes. III. Acta Sci. Math.(Szeged) 14 (1951), 69–82.Google Scholar
Przytycki, P. & Wise, D.. Mixed 3-manifolds are virtually special. J. Amer. Math. Soc. 31.2 (2018), 319–347.10.1090/jams/886CrossRefGoogle Scholar
Servatius, H., Droms, C. & Servatius, B.. Surface subgroups of graph groups. Proc. Amer. Math. Soc. 106.3 (1989), 573–578.CrossRefGoogle Scholar
Wise, D.. The structure of groups with a quasiconvex hierarchy. Preprint (2011).Google Scholar