Essentially, the theory of Vassiliev invariants of braids is a particular case of the Vassiliev theory for tangles, and the main constructions are very similar to the case of knots. There is, however, one big difference: many of the questions that are still open for knots are rather easy to answer in the case of braids. This, in part, can be explained by the fact that braids form groups, and it turns out that the whole Vassiliev theory for braids can be described in group-theoretic terms. In this chapter we shall see that the Vassiliev filtration on the pure braid groups coincides with the filtrations coming from the nilpotency theory of groups. In fact, for any given group the nilpotency theory could be thought of as a theory of finite type invariants.

The group-theoretic techniques of this chapter can be used to study knots and links. One such application is the theorem of Goussarov which says that n-equivalence classes of string links on m strands form a group. Another application of the same methods is a proof that Vassiliev invariants of pure braids extend to invariants of string links of the same order. In order to make these connections, we shall describe a certain braid closure that produces string links out of pure braids.

The theory of the finite type invariants for braids was first developed by T. Kohno (1985, 1987) several years before Vassiliev knot invariants were introduced.