Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-24T21:58:06.422Z Has data issue: false hasContentIssue false
  • English
  • Français

CONGRUENCE SUBGROUPS OF BRAID GROUPS AND CRYSTALLOGRAPHIC QUOTIENTS. PART I

Part of: Other groups of matrices Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 September 2024

PAOLO BELLINGERI ,
CELESTE DAMIANI Open the ORCID record for CELESTE DAMIANI [Opens in a new window] ,
OSCAR OCAMPO Open the ORCID record for OSCAR OCAMPO [Opens in a new window]  and
CHARALAMPOS STYLIANAKIS
PAOLO BELLINGERI*
Affiliation:
Normandie Université, UNICAEN, CNRS, LMNO, 14000 Caen, France
CELESTE DAMIANI
Affiliation:
Fondazione Istituto Italiano di Tecnologia, Genova, Italy e-mail: [email protected]
OSCAR OCAMPO
Affiliation:
Universidade Federal da Bahia, Departamento de Matemática – IME, CEP: 40170-110 Salvador, BA, Brazil e-mail: [email protected]
CHARALAMPOS STYLIANAKIS
Affiliation:
University of the Aegean, Department of Mathematics, Karlovasi 83200, Samos, Greece e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

Keywords

braid groupsmapping class groupscongruence subgroupssymplectic representation

MSC classification

Primary: 20F36: Braid groups; Artin groups
Secondary: 20H15: Other geometric groups, including crystallographic groups 20F65: Geometric group theory 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 118 , Issue 2 , April 2025 , pp. 145 - 163
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

Footnotes

The first author was partially supported by the ANR project AlMaRe (ANR-19-CE40-0001). The second author is a member of GNSAGA of INdAM, and was partially supported Leverhulme Trust research project grant ‘RPG-2018-029: Emergent Physics From Lattice Models of Higher Gauge Theory’. The third author would like to thank Laboratoire de Mathématiques Nicolas Oresme (Université de Caen Normandie) for their hospitality from August 2023 to January 2024, where part of this project was developed, and was partially supported by Capes/Programa Capes-Print/ Processo número 88887.835402/2023-00 and also by National Council for Scientific and Technological Development – CNPq through a Bolsa de Produtividade 305422/2022-7.

Communicated by Ben Martin

References

Appel, J., Bloomquist, W., Gravel, K. and Holden, A., ‘On quotients of congruence subgroups of braid groups’, Preprint, 2020, arXiv:2011.13876.Google Scholar
A’Campo, N., ‘Tresses, monodromie et le groupe symplectique’, Comment. Math. Helv. 54(2) (1979), 318–327.CrossRefGoogle Scholar
Arnol’d, V. I., ‘A remark on the branching of hyperelliptic integrals as functions of the parameters’, Funkcional. Anal. i Priložen. 2(3) (1968), 1–3.Google Scholar
Artin, E., ‘Theorie der Zöpfe’, Abh. Math. Semin. Univ. Hambg. 4 (1925), 47–72.CrossRefGoogle Scholar
Birman, J. S. and Brendle, T. E., ‘Braids: a survey’, in: Handbook of Knot Theory (Elsevier B. V., Amsterdam, 2005), 19–103.CrossRefGoogle Scholar
Bharathram, V. and Birman, J., ‘On the Burau representation of B4’, Involve 14(1) (2021), 143–154.CrossRefGoogle Scholar
Bellingeri, P., Damiani, C., Ocampo, O. and Stylianakis, C., ‘Congruence subgroups of braid groups and crystallographic quotients. Part II’, Preprint, 2024, arXiv:2404.05804.Google Scholar
Bellingeri, P., Guaschi, J. and Makri, S., ‘Unrestricted virtual braids and crystallographic braid groups’, Bol. Soc. Mat. Mex. (3) 28(3) (2022), Article no. 63, 16 pages.Google Scholar
Bigelow, S. J., ‘The Burau representation is not faithful for $n=5$ ’, Geom. Topol. 3 (1999), 397–404.CrossRefGoogle Scholar
Bigelow, S. J., ‘Braid groups are linear’, J. Amer. Math. Soc. 14(2) (2001), 471–486.CrossRefGoogle Scholar
Bell, R. W. and Margalit, D., ‘Braid groups and the co-Hopfian property’, J. Algebra 303(1) (2006), 275–294.Google Scholar
Brendle, T. E. and Margalit, D., ‘The level four braid group’, J. reine angew. Math. 735 (2018), 249–264.CrossRefGoogle Scholar
Beck, V. and Marin, I., ‘Torsion subgroups of quasi-abelianized braid groups’, J. Algebra 558 (2020), 3–23.CrossRefGoogle Scholar
Brendle, T. E., Margalit, D. and Putman, A., ‘Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at $t=-1$ ’, Invent. Math. 200(1) (2015), 263–310.Google Scholar
Bass, H., Milnor, J. and Serre, J.-P., ‘Solution of the congruence subgroup problem for $s{l}_n\left(n\ge 3\right)$ and $s{p}_{2n}\left(n\ge 2\right)$ ’, Publ. Math. Inst. Hautes Études Sci. 33 (1967), 59–137.CrossRefGoogle Scholar
Bloomquist, W., Patzt, P. and Scherich, N., ‘Quotients of braid groups by their congruence subgroups,’ Proc. Amer. Math. Soc., to appear.Google Scholar
Beridze, A. and Traczyk, P., ‘Burau representation for $n=4$ ’, J. Knot Theory Ramifications 27(3) (2018), Article no. 1840002, 6 pages.CrossRefGoogle Scholar
Burau, W., ‘Über zopfgruppen und gleichsinnig verdrillte verkettungen’, Abh. Math. Semin. Univ. Hambg. 11 (1935), 179–186.Google Scholar
Cerqueira dos Santos Júnior, P. C. and Ocampo, O., ‘Virtual braid groups, virtual twin groups and crystallographic groups’, J. Algebra 632 (2023), 567–601.CrossRefGoogle Scholar
Chow, W.-L., ‘On the algebraical braid group’, Ann. of Math. (2) 49 (1948), 654–658.Google Scholar
Coxeter, H. S. M., ‘Factor groups of the braid group’, in: Proceedings of the Fourth Canadian Mathematical Congress (Banff, 1957) (University of Toronto Press, 1959), 95–122.Google Scholar
Datta, A., ‘A strong characterization of the entries of the Burau matrices of $4$ -braids: The Burau representation of the braid group ${B}_4$ is faithful almost everywhere’, Preprint, 2022, arXiv:2209.10826.Google Scholar
Dekimpe, K., Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathematics, 1639 (Springer-Verlag, Berlin, 1996).Google Scholar
Farb, B. and Margalit, D., A Primer on Mapping Class Groups, Princeton Mathematical Series, 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Gambaudo, J.-M. and Ghys, É., ‘Braids and signatures’, Bull. Soc. Math. France 133(4) (2005), 541–579.Google Scholar
Gonçalves, D. L., Guaschi, J. and Ocampo, O., ‘A quotient of the Artin braid groups related to crystallographic groups’, J. Algebra 474 (2017), 393–423.Google Scholar
Gonçalves, D. L., Guaschi, J., Ocampo, O. and Pereiro, C., ‘Crystallographic groups and flat manifolds from surface braid groups’, Topology Appl. 293 (2021), Article no. 107560, 16 pages.Google Scholar
Johnson, D. L., Presentations of Groups, London Mathematical Society Student Texts, 15 (Cambridge University Press, Cambridge, 1990).Google Scholar
Kordek, K. and Margalit, D., ‘Representation stability in the level 4 braid group’, Math. Z. 302 (2022), 433–482.CrossRefGoogle Scholar
Kumar, P., Naik, T. K. and Singh, M., ‘Congruence subgroups and crystallographic quotients of small Coxeter groups’, Forum Math. 36(1) (2024), 193–213.CrossRefGoogle Scholar
Krammer, D., ‘Braid groups are linear’, Ann. of Math. (2) 155(1) (2002), 131–156.CrossRefGoogle Scholar
Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, 247 (Springer, New York, 2008).CrossRefGoogle Scholar
Long, D. D. and Paton, M., ‘The Burau representation is not faithful for $n\ge 6$ ’, Topology 32(2) (1993), 439–447.CrossRefGoogle Scholar
Murasugi, K. and Kurpita, B. I., A Study of Braids, Mathematics and Its Applications, 484 (Kluwer Academic Publishers, Dordrecht, 1999).Google Scholar
Moody, J. A., ‘The Burau representation of the braid group ${B}_n$ is unfaithful for large $n$ ’, Bull. Amer. Math. Soc. (N.S.) 25(2) (1991), 379–384.CrossRefGoogle Scholar
Magnus, W. and Peluso, A., ‘On a theorem of V. I. Arnol’d’, Comm. Pure Appl. Math. 22 (1969), 683–692.CrossRefGoogle Scholar
Nakamura, T. ‘The cohomology class of the mod 4 braid group’, Int. J. Algebra Comput. 32 (2021), 1125–1159.Google Scholar
Perron, B. and Vannier, J. P., ‘Groupe de monodromie géométrique des singularités simples’, Math. Ann. 306(1) (1996), 231–245.Google Scholar
Raghunathan, M. S., ‘The congruence subgroup problem’, Proc. Math. Sci. 114 (2004), 299–308.Google Scholar
Stylianakis, C., ‘Congruence subgroups of braid groups’, Internat. J. Algebra Comput. 28(2) (2018), 345–364.CrossRefGoogle Scholar
Tits, J., ‘Normalisateurs de tores i: Groupes de coxeter étendus’, J. Algebra 4 (1966), 96–116.CrossRefGoogle Scholar