Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T06:39:38.133Z Has data issue: false hasContentIssue false
  • English
  • Français

ABELIAN AND METABELIAN QUOTIENT GROUPS OF SURFACE BRAID GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  10 June 2016

PAOLO BELLINGERI ,
EDDY GODELLE  and
JOHN GUASCHI
PAOLO BELLINGERI
Affiliation:
Normandie Univ, France, e-mails: paolo.bellingeri@unicaen.fr, eddy.godelle@unicaen.fr, john.guaschi@unicaen.fr UNICAEN, LMNO, CNRS UMR 6139, F-14032 Caen, France
EDDY GODELLE
Affiliation:
Normandie Univ, France, e-mails: paolo.bellingeri@unicaen.fr, eddy.godelle@unicaen.fr, john.guaschi@unicaen.fr UNICAEN, LMNO, CNRS UMR 6139, F-14032 Caen, France
JOHN GUASCHI
Affiliation:
Normandie Univ, France, e-mails: paolo.bellingeri@unicaen.fr, eddy.godelle@unicaen.fr, john.guaschi@unicaen.fr UNICAEN, LMNO, CNRS UMR 6139, F-14032 Caen, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study Abelian and metabelian quotients of braid groups of oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell–Neuwirth fibrations.

MSC classification

Primary: 20F14: Derived series, central series, and generalizations
Secondary: 20F36: Braid groups; Artin groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 119 - 142
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. An, B. H. and Ko, K. H., A family of representations of braid groups on surfaces, Pacific J. Math. 247 (2010), 257–282.CrossRefGoogle Scholar
2. Bardakov, V. G., Extending representations of braid groups to the automorphism groups of free groups, J. Knot Theory Ramifications 14 (2005), 1087–1098.CrossRefGoogle Scholar
3. Bardakov, V. G., Linear representations of the braid groups of some manifolds, Acta Appl. Math. 85 (2005), 41–48.CrossRefGoogle Scholar
4. Bellingeri, P., On presentations of surface braid groups, J. Algebra 274 (2004), 543–563.CrossRefGoogle Scholar
5. Bellingeri, P. and Funar, L., Braids on surfaces and finite type invariants, C. R. Math. Acad. Sci. Paris 338 (2004), 157–162.CrossRefGoogle Scholar
6. 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
7. Bigelow, S., Braid groups are linear, J. Am. Math. Soc. 14 (2001), 471–486.CrossRefGoogle Scholar
8. Bigelow, S. and Budney, R., The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001), 699–708.CrossRefGoogle Scholar
9. Birman, J. S., Braids, links, and mapping class groups, Ann. Math. Stud., vol. 82, (Princeton Univ. Press, Princeton, NJ, 1974).Google Scholar
10. Cohen, A. and Wales, D. B., Linearity of Artin groups of finite type, Israel J. Math. 131 (2002), 101–123.CrossRefGoogle Scholar
11. Digne, F., On the linearity of Artin braid groups, J. Algebra 268 (2003), 39–57.CrossRefGoogle Scholar
12. Fadell, E. and Neuwirth, L., Configuration spaces, Math. Scand. 10 (1962), 111–118.CrossRefGoogle Scholar
13. Fadell, E. and Van Buskirk, J., The braid groups of E 2 and S 2 , Duke Math. J. 29 (1962), 243–257.CrossRefGoogle Scholar
14. Fenn, R., Rolfsen, D. and Zhu, J., Centralisers in the braid group and singular braid monoid, Enseign. Math. 42 (1996), 75–96.Google Scholar
15. Fox, R. H. and Neuwirth, L., The braid groups, Math. Scand. 10 (1962), 119–126.CrossRefGoogle Scholar
16. 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
17. 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
18. Gonçalves, D. L. and Guaschi, J., The braid group $B_{n,m}(\mathbb{S}^2)$ and a generalisation of the Fadell-Neuwirth short exact sequence, J. Knot Theory Ramifications 14 (2005), 375–403.CrossRefGoogle Scholar
19. Gonçalves, D. L. and Guaschi, J., The lower central and derived series of the braid groups of the finitely-punctured sphere, J. Knot Theory Ramifications 18 (2009), 651–704.CrossRefGoogle Scholar
20. Gonçalves, D. L. and Guaschi, J., The Fadell-Neuwirth short exact sequence for non-orientable surfaces, J. Pure Appl. Algebra 214 (2010), 667–677.CrossRefGoogle Scholar
21. Gonçalves, D. L. and Guaschi, J., Surface braid groups and coverings, J. London Math. Soc. 85 (2012), 855–868.CrossRefGoogle Scholar
22. Häring-Oldenburg, R. and Lambropoulou, S., Knot theory in handlebodies, in Knots 2000 Korea, Vol. 3 (Yongpyong), J. Knot Theory Ramifications 11 (2002), 921–943.CrossRefGoogle Scholar
23. Johnson, D. L., Presentation of groups, LMS Lectures Notes 22 (Cambridge University Press, New York/Melbourne, 1976).Google Scholar
24. Kassel, C. and Turaev, V., Braid groups, Graduate Texts in Mathematics 247 (Springer, New York, 2008).Google Scholar
25. Krammer, D., Braid groups are linear, Ann. Math. 155 (2002), 131–156.CrossRefGoogle Scholar
26. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, 2nd edition (Dover Publications Inc., Mineola, NY, 2004).Google Scholar
27. Manfredini, S., Some subgroups of Artin's braid group, Topology Appl. 78 (1997), 123–142.CrossRefGoogle Scholar
28. Paris, L. and Rolfsen, D., Geometric subgroups of surface braid groups, Ann. Inst. Fourier 49 (1999), 417–472.CrossRefGoogle Scholar
29. Zheng, H., Faithfulness of the Lawrence representation of braid group, arXiv:math/0509074.Google Scholar