Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-20T01:47:08.773Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower central series, surface braid groups, surjections and permutations

Part of: Special aspects of infinite or finite groups Permutation groups

Published online by Cambridge University Press:  05 April 2021

PAOLO BELLINGERI ,
DACIBERG LIMA GONÇALVES  and
JOHN GUASCHI
PAOLO BELLINGERI
Affiliation:
Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14000 Caen, France. e-mail: paolo.bellingeri@unicaen.fr
DACIBERG LIMA GONÇALVES
Affiliation:
Departamento de Matemática - IME-USP, Rua do Matão 1010 CEP: 05508-090 - São Paulo - SP - Brazil. e-mail: dlgoncal@ime.usp.br
JOHN GUASCHI
Affiliation:
Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14000 Caen, France. e-mail: john.guaschi@unicaen.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

MSC classification

Primary: 20F36: Braid groups; Artin groups
Secondary: 20F14: Derived series, central series, and generalizations 20B15: Primitive groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 373 - 399
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical 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.)

References

An, B. H.. Automorphisms of braid groups on orientable surfaces. J. Knot Theory Ramifications 25 (2016), 1650022, 32 pp.CrossRefGoogle Scholar
Aramayona, J., Leininger, C. and Souto, J.. Injections of mapping class groups. Geom. Topol. 13 (2009), 2523–2541.CrossRefGoogle Scholar
Aramayona, J. and Souto, J.. Homomorphisms between mapping class groups. Geom. Topol. 16 (2012), 2285–2341.CrossRefGoogle Scholar
Artin, E.. Theory of braids. Ann. Math. 48 (1947), 101–126.CrossRefGoogle Scholar
Artin, E.. Braids and permutations. Ann. Math. 48 (1947), 643–649.CrossRefGoogle Scholar
Bardakov, V. and Bellingeri, P.. On residual properties of pure braid groups of closed surfaces. Comm. Algebra 37 (2009), 1481–1490.CrossRefGoogle Scholar
Bardakov, V., Neshchadim, M. and Singh, M.. Automorphisms of pure braid groups. Monatsh. Math. 187 (2019), 1–19.CrossRefGoogle Scholar
Bell, R. and Margalit, D.. Braid groups and the co-Hopfian property. J. Algebra 303 (2006), 275–294.CrossRefGoogle Scholar
Bellingeri, P.. On presentations of surface braid groups. J. Algebra 274 (2004), 543–563.CrossRefGoogle Scholar
Bellingeri, P.. On automorphisms of surface braid groups. J. Knot Theory Ramifications 17 (2008) 1–11.CrossRefGoogle Scholar
Bellingeri, P. and Gervais, S.. On p-almost direct products and residual properties of pure braid groups of non-orientable surfaces. Algebra Geom. Topol. 16 (2016), 547–568.CrossRefGoogle Scholar
Bellingeri, P., Gervais, S. and Guaschi, J.. Lower central series of Artin–Tits and surface braid groups. J. Algebra 319 (2008), 1409–1427.CrossRefGoogle Scholar
Bellingeri, P., Godelle, E. and Guaschi, J.. Abelian and metabelian quotients of surface braid groups. Glasgow J. Math. 59 (2017), 119–142.CrossRefGoogle Scholar
Berrick, A. J., Gebhardt, V. and Paris, L.. Finite index subgroups of mapping class groups. Proc. London Math. Soc. 108 (2014), 575–599.Google Scholar
Castel, F.. Geometric representations of the braid groups. Astérisque 378 (2016).Google Scholar
Chen, L.. Surjective homomorphisms between surface braid groups. Israel J. Math. 232 (2019), 483–500.CrossRefGoogle Scholar
Chen, L., Kordek, K. and Margalit, D.. Homomorphisms between braid groups, arXiv:1910.00712.Google Scholar
Chen, L. and Mukherejea, A.. From braid groups to mapping class groups. arXiv:2011.13020.Google Scholar
Chudnovsky, A., Kordek, K., Li, Q. and Partin, C.. Finite quotients of braid groups. arXiv:1910.07177, Geom. Dedicata, to appear.Google Scholar
Dyer, J. L. and Grossman, E. K.. The automorphism groups of the braid groups. Amer. J. Math. 103 (1981), 1151–1169.CrossRefGoogle Scholar
Fadell, E. and Van Buskirk, J.. The braid groups of E 2 and S 2. Duke Math. J. 29 (1962), 243–257.CrossRefGoogle Scholar
Falk, M. and Randell, R.. The lower central series of a fiber-type arrangement. Invent. Math. 82 (1985), 77–88.CrossRefGoogle Scholar
Fox, R. H. and Neuwirth, L.. The braid groups. Math. Scand. 10 (1962), 119–126.CrossRefGoogle Scholar
Gaglione, A. M.. Factor groups of the lower central series for special free products. J. Algebra 37 (1975), 172–185.CrossRefGoogle Scholar
Gillette, R. and Van Buskirk, J.. The word problem and consequences for the braid groups and mapping class groups of the 2-sphere. Trans. Amer. Math. Soc. 131 (1968), 277–296.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. On the structure of surface pure braid groups. J. Pure Appl. Algebra 182 (2003), 33–64 (due to a printer’s error, this article was republished in its entirety with the reference 186 (2004), 187–218).CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The roots of the full twist for surface braid groups. Math. Proc. Camb. Phil. Soc. 137 (2004), 307–320.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The braid groups of the projective plane. Algebr. Geom. Topol. 4 (2004), 757–780.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The braid group Bn,m (S2) and a generalisation of the Fadell–Neuwirth short exact sequence. J. Knot Theory Ramifications 14 (2005), 375–403.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The quaternion group as a subgroup of the sphere braid groups. Bull. London Math. Soc. 39 (2007), 232–234CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The lower central and derived series of the braid groups of the sphere. Trans. Amer. Math. Soc. 361 (2009), 3375–3399.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence. J. Pure Appl. Algebra 214 (2010), 667–677.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. The lower central and derived series of the braid groups of the projective plane. J. Algebra 331 (2011), 96–129.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. Surface braid groups and coverings. J. London Math. Soc. 85 (2012), 855–868.CrossRefGoogle Scholar
Gonçalves, D. L. and Guaschi, J.. Minimal generating and normally generating sets for the braid and mapping class groups of D2, S2 and RP 2. Math. Z. 274 (2013), 667–683.CrossRefGoogle Scholar
Gruenberg, K. W.. Residual properties of infinite soluble groups. Proc. London Math. Soc. 7 (1957), 29–62.CrossRefGoogle Scholar
Guaschi, J. and de Miranda, C. e Pereiro. Lower central and derived series of semi-direct products, and applications to surface braid groups. J. Pure Appl. Algebra 224 (2020), 106309.CrossRefGoogle Scholar
Irmak, E., Ivanov, N. V. and McCarthy, J. D.. Automorphisms of surface braid groups, arXiv:math/0306069.Google Scholar
Ivanov, N. V.. Permutation representations of braid groups of surfaces. Math. USSR-Sb. 71 (1992), 309–318.CrossRefGoogle Scholar
Ivanov, N. V.. Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 14 (1997), 651–666.CrossRefGoogle Scholar
Kohno, T.. Série de Poincaré–Koszul associée aux groupes de tresses pures. Invent. Math. 82 (1985), 57–75.Google Scholar
Labute, J.. On the descending central series of groups with a single defining relation. J. Algebra 14 (1970), 16–23.CrossRefGoogle Scholar
Ya, V.. Lin. Representation of a braid group by permutations. Uspekhi Mat. Nauk 27 (1972), 165–192.Google Scholar
Ya, V.. Lin. Braids and permutations, arXiv:math/0404528.Google Scholar
Magnus, W., Karrass, A. and Solitar, D.. Combinatorial Group Theory, reprint of the 1976 second edition (Dover Publications, Inc., Mineola, NY, 2004).Google Scholar
de Miranda, C. e Pereiro. Os grupos de tranças do toro e da garrafa de Klein. Ph.D thesis, Universidade Federal de São Carlos, Brazil, and Université de Caen Basse-Normandie, France (2015).Google Scholar
Murasugi, K.. Seifert fibre spaces and braid groups. Proc. London Math. Soc 44 (1982), 71–84.CrossRefGoogle Scholar
Paris, L. and Rolfsen, D.. Geometric subgroups of surface braid groups. Ann. Inst. Fourier 49 (1999), 417–472.CrossRefGoogle Scholar
Paris, L. and Rolfsen, D.. Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521 (2000), 47–83.Google Scholar
Shackleton, K. J.. Combinatorial rigidity in curve complexes and mapping class groups. Pac. J. Math. 230 (2007), 217–232.CrossRefGoogle Scholar
Van Buskirk, J.. Braid groups of compact 2-manifolds with elements of finite order. Trans. Amer. Math. Soc. 122 (1966), 81–97.CrossRefGoogle Scholar