Stephen Bigelow
Published 1999-04-20, updated 1999-11-30Version 2
The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton improved on Moody's techniques to bring this down to n>5. Their construction uses a simple closed curve on the 6-punctured disc with certain homological properties. In this paper we give such a curve on the 5-punctured disc, thus proving that the Burau representation is not faithful for n>4.
Comments: 8 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper16.abs.html
Journal: Geom. Topol. 3 (1999), 397-404
On the Burau representation of $B_3$
Braid groups are almost co-Hopfian
On the Burau representation for $n=4$
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