The Burau representation is not faithful for n = 5

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.Comment: 8 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper16.abs.htm

A representation of generalized braid group in classical braid group

We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type A_n, B_n, I_2(k). The injectivity question of this homomorphism (in these particular cases) is not yet settled. If this map is an injective homomorphism then several results will follow. For example it will follow that the Whitehead group, projective class group and the lower K-group of any subgroup of any finite type generalized braid group vanish. (For the classical braid group case this vanishing result is proved by the author and F.T. Farrell in the paper "The Whitehead groups of braid groups vanish".) Also it will follow that a finite type generalized braid group satisfies Tits alternative (recently this was asked by M. Bestvina).Comment: 22 pages, 12 figures, 1 table. it is a zipped file containing all the figure files in postscript format and the amstex source of the articl

Cabling Burau Representation

The Burau representation enables to define many other representations of the braid group $B_n$ by the topological operation of ``cabling braids''. We show here that these representations split into copies of the Burau representation itself and of a representation of $B_n/(P_n,P_n)$. In particular, we show that there is no gain in terms of faithfulness by cabling the Burau representation.Comment: 11 page