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The level four braid group

Tara E. Brendle EMAIL logo and Dan Margalit


By evaluating the Burau representation at t= -1, one obtains a symplectic representation of the braid group. We study the resulting congruence subgroups of the braid group, namely, the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup is equal to the group generated by squares of Dehn twists. We also show that the image of the Brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup.

Award Identifier / Grant number: EP/J019593/1

Award Identifier / Grant number: DMS-1057874

Funding statement: The first author is supported in part by EPSRC grant EP/J019593/1. The second author supported by the National Science Foundation under Grant No. DMS-1057874.


We would like to thank Joan Birman, Neil Fullarton, Louis Funar, Lalit Jain, Joseph Rabinoff, Douglas Ulmer, Kirsten Wickelgren, and Mante Zelvyte for helpful conversations. We would especially like to thank Andrew Putman, who pointed out a mistake in an earlier version and made several useful comments. We are also grateful to Jordan Ellenberg; a conversation with him on MathOverflow partly inspired the work in the last section.


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Received: 2014-11-22
Revised: 2015-2-17
Published Online: 2015-7-31
Published in Print: 2018-2-1

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