Braid Groups Are Linear

The braid groups Bn were originally introduced by Emil Artin in 1926. They have many interpretations, for example, as the group of geometric braids in R, as the Artin group of type An, as the fundamental group of a certain hyperplane arrangement, as a subgroup of the automorphism group of a free group, and so on. In this paper we will use the interpretation of Bn as the mapping class group of an n-times punctured disk. A group is said to be linear if it admits a faithful representation into GL(m,R) for some natural number m. The question of whether braid groups are linear probably dates back to 1935 when Burau [Bur36] discovered an n-dimensional representation of Bn. For a long time this was thought to be a possible candidate for a faithful representation. A simple proof that it is faithful in the case n = 3 gave some reason for optimism. However in 1991, Moody [Moo91] showed that the Burau representation is not faithful for n ≥ 9. This was later brought down to n ≥ 6 and then n ≥ 5 in the papers [LP93] and [Big99]. It therefore came as a pleasant surprise when Krammer [Kra99] proved that another representation of the braid groups is faithful in the case n = 4. The representation Krammer used is essentially the same as one used by Lawrence in [Law90] to give a topological definition of a certain summand of the Jones representation. We call this representation the Lawrence-Krammer representation. In this paper, we prove the following.