Advertisement

Geometriae Dedicata

, Volume 169, Issue 1, pp 145–163 | Cite as

On Burau’s representations at roots of unity

  • Louis Funar
  • Toshitake Kohno
Original Paper

Abstract

We consider subgroups of the braid groups which are generated by \(n\)th powers of the standard generators and prove that any infinite intersection (with even \(n\)) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau’s representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.

Keywords

Mapping class group Dehn twist Temperley–Lieb algebra Triangle group Braid group Burau representation 

Mathematics Subject Classification (2000)

57 M 07 20 F 36 20 F 38 57 N 05 

Notes

Acknowledgments

We are grateful to Norbert A’Campo, Jørgen Andersen, Jean-Benoît Bost, Martin Deraux, Greg Kuperberg, François Labourie, Yves Laszlo, Greg McShane, Ivan Marin, Gregor Masbaum, Daniel Matei, Majid Narimannejad, Christian Pauly, Bob Penner, Christophe Sorger and Richard Wentworth for useful discussions and to the referees for a careful reading of the paper leading to numerous corrections and suggestions. The first author was partially supported by ANR-06-BLAN-0311 Repsurf and ANR 2011 BS 01 020 01 ModGroup. The second author is partially supported by Grant-in-Aid for Scientific Research 20340010, Japan Society for Promotion of Science, and by World Premier International Research Center Initiative, MEXT, Japan. A part of this work was accomplished while the second author was staying at Institut Fourier in Grenoble. He would like to thank Institut Fourier for hospitality.

References

  1. 1.
    Andersen, J.E.: Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups. Ann. Math. 163, 347–368 (2006)CrossRefMATHGoogle Scholar
  2. 2.
    Andersen, J.E., Masbaum, G., Ueno, K.: Topological quantum field theory and the Nielsen-Thurston classification of M(0,4). Math. Proc. Camb. Phil. Soc. 141, 477–488 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beardon, A.F.: The geometry of discrete groups. Corrected reprint of the 1983 original. Graduate Texts in Mathematics, 91, Springer, New York (1995)Google Scholar
  4. 4.
    Bigelow, S.: The Burau representation is not faithful for n = 5. Geom. Topol. 3, 397–404 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Topological quantum field theories derived from the Kauffman bracket. Topology 34, 883–927 (1995)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Church, T., Farb, B.: Infinite generation of the kernels of the Magnus and Burau representations. Algebr. Geom. Topol. 10, 837–851 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cooper, D., Long, D.D.: A presentation for the image of Burau(4) \(\otimes {\mathbb{Z}}_2\). Invent. Math. 127, 535–570 (1997)Google Scholar
  8. 8.
    Coxeter, H.S.M.: On factors of braid groups. In: Proceedings of 4-th Canadian Math. Congress, Banff: University of Toronto Press, vol. 1959, pp. 95–122 (1957)Google Scholar
  9. 9.
    Coxeter, H.S.M.: Regular Complex Polytopes. Cambridge University Press, Cambridge (1974)MATHGoogle Scholar
  10. 10.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and relations for discrete groups, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14. Springer, Berlin, pp. ix+169 (1980)Google Scholar
  11. 11.
    Deraux, M.: On the universal cover of certain exotic Kähler surfaces of negative curvature. Math. Annalen 329, 653–683 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Freedman, M.H., Walker, K., Wang, Z.: Quantum SU(2) faithfully detects mapping class groups modulo center. Geom. Topol. 6, 523–539 (2002)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Funar, L., Kohno, T.: Free subgroups within the images of quantum representations. 19 p., Forum Math. (to appear) arxiv:1108.4904Google Scholar
  14. 14.
    Gilmer, P., Masbaum, G.: Integral TQFT for a one-holed torus. Pac. J. Math. 252, 93–112 (2011)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jones, V.F.R.: Braid groups, Hecke algebras and type \(II_1\) factors. In: Araki, H., Effros, E.G. (eds.) Geometric Methods in Operator Algebras, US-Japan Seminar, Proc, pp. 242–271 (1986)Google Scholar
  16. 16.
    Knapp, A.W.: Doubly generated Fuchsian groups. Mich. J. Math. 15, 289–304 (1968)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Larsen, M., Zhenghan, Wang: Density of the SO(3) TQFT representation of mapping class groups. Commun. Math. Phys. 260, 641–658 (2005)CrossRefMATHGoogle Scholar
  18. 18.
    Long, D.D., Paton, M.: The Burau representation is not faithful for \(n\ge 6\). Topology 32, 439–447 (1993)Google Scholar
  19. 19.
    Masbaum, G.: On representations of mapping class groups in integral TQFT. Oberwolfach Rep. 5(2), 1202–1205 (2008)Google Scholar
  20. 20.
    McMullen, C.T.: Braid groups and Hodge theory. Math. Ann. 355, 893–946 (2013)Google Scholar
  21. 21.
    Moody, J.A.: The faithfulness question for the Burau representation. Proc. Am. Math. Soc. 119(2), 671–679 (1993)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Mostow, G.D.: On discontinuous action of monodromy groups on the complex n-balls. J. Am. Math. Soc. 1, 555–586 (1988)MATHMathSciNetGoogle Scholar
  23. 23.
    Paris, L., Rolfsen, D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)MATHMathSciNetGoogle Scholar
  24. 24.
    Radin, C., Sadun, L.: On 2-generator subgroups of SO(3). Trans. Am. Math. Soc. 351, 4469–4480 (1999)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Santharoubane, R.: Limits of the quantum SO(3) representations for the one-holed torus. arXiv:1202.1813Google Scholar
  26. 26.
    Squier, C.: The Burau representation is unitary. Proc. Am. Math. Soc. 90, 199–202 (1984)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Wenzl, H.: On sequences of projections. CR Math. Rep. Acad. Sci. Can. 9, 5–9 (1987)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut Fourier BP 74, UMR 5582University of Grenoble ISaint-Martin-d’Hères CedexFrance
  2. 2.Kavli IPMU, Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations