Geometriae Dedicata

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

On Burau’s representations at roots of unity

  • Louis FunarEmail author
  • Toshitake Kohno
Original Paper


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.


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 



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.


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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

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