Skip to main content
SpringerLink
Log in

Linear-central filtrations and the image of the Burau representation

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give an approximate answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative to a topology induced by a naturally-occurring filtration. We expect that the methods of the paper should extend to many other representations of the braid group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. To see this, observe that tI is central in \(\Gamma \), but that the center of \(B_3\) is sent to the cyclic group \(\langle t^3 I \rangle \). Thus if \(tI = \beta (b)\) for some \(b \in B_3\), this b must necessarily be non-central, but this contradicts the fact that \(\beta \) is faithful for \(n = 3\).

References

  1. Bigelow, S.: The Burau representation is not faithful for \(n=5\). Geom. Topol. 3, 397–404 (1999)

    Article  MathSciNet  Google Scholar 

  2. Birman, J.S.: Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton (1994)

    Google Scholar 

  3. Bass, H., Lubotzky, A.: Linear-central filtrations on groups. In: Abikoff, W., Birman, J.S., Kuiken, K. (eds.) The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions. Volume 169 of Contemporary Mathematics, pp. 45–98. American Mathematical Society, Providence (1994)

    Chapter  Google Scholar 

  4. Brendle, T., Margalit, D., Putman, A.: Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at \(t=-1\). Invent. Math. 200(1), 263–310 (2015)

    Article  MathSciNet  Google Scholar 

  5. Burau, W.: Über Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Sem. Univ. Hamburg 11(1), 179–186 (1935)

    Article  MathSciNet  Google Scholar 

  6. Church, T., Farb, B.: Infinite generation of the kernels of the Magnus and Burau representations. Algebr. Geom. Topol. 10(2), 837–851 (2010)

    Article  MathSciNet  Google Scholar 

  7. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Volume 49 of Princeton Mathematical Series. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  8. Griffiths, P., Schmid, W.: Recent developments in Hodge theory: a discussion of techniques and results. Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), pp. 31–127 (1975)

  9. Hain, R.: Finiteness and torelli spaces. In: Problems on mapping class groups and related topics. In: Volume 74 of Proceeding of Symposium on Pure Mathematics, pp. 57–70. American Mathematics Society, Providence (2006)

  10. Kordek, K., Salter, N.: An abelian quotient of the braid Torelli group. In preparation (2019)

  11. Long, D.D.: On the linear representation of braid groups. Trans. Am. Math. Soc. 311(2), 535–560 (1989)

    Article  MathSciNet  Google Scholar 

  12. Long, D.D., Paton, M.: The Burau representation is not faithful for \(n\ge 6\). Topology 32(2), 439–447 (1993)

    Article  MathSciNet  Google Scholar 

  13. Lee, R., Szczarba, R.H.: On the homology and cohomology of congruence subgroups. Invent. Math. 33(1), 15–53 (1976)

    Article  MathSciNet  Google Scholar 

  14. McMullen, C.T.: Braid groups and Hodge theory. Math. Ann. 355(3), 893–946 (2013)

    Article  MathSciNet  Google Scholar 

  15. Moody, J.A.: The Burau representation of the braid group \(B_n\) is unfaithful for large \(n\). Bull. Am. Math. Soc. (N.S.) 25(2), 379–384 (1991)

    Article  Google Scholar 

  16. Seifert, H.: Über das Geschlecht von Knoten. Math. Ann. 110(1), 571–592 (1935)

    Article  MathSciNet  Google Scholar 

  17. Squier, C.C.: The Burau representation is unitary. Proc. Am. Math. Soc. 90(2), 199–202 (1984)

    Article  MathSciNet  Google Scholar 

  18. Venkataramana, T.N.: Image of the Burau representation at \(d\)-th roots of unity. Ann. Math. (2) 179(3), 1041–1083 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Kevin Kordek for many useful discussions and for collaboration on some early results in the direction of Theorem A. I am also indebted to Alex Suciu for pointing out the connection between the s-adic filtration and the augmentation ideal of \(\Lambda \). I thank the referee for their careful attention to the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Nick Salter

Authors
  1. Nick Salter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nick Salter.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based upon work supported by the National Science Foundation under Award No. DMS-1703181.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salter, N. Linear-central filtrations and the image of the Burau representation. Geom Dedicata 211, 145–163 (2021). https://doi.org/10.1007/s10711-020-00544-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-020-00544-4

Keywords

Search

Navigation