Braid subgroup normalisers, commensurators and induced representations

Abstract

The classical braid groups of E. Artin form an ascending sequence B1 B2 . . . B1. These groups, and their representations, play an important roA le in several areas of mathematics and physics. The object of this note is to establish new algebraic information about this sequence of groups. In particular, we determine the normaliser and commensurator of the braid group Bn in Bm, n m 1. They are described in terms of the centraliser, which was recently characterised in [FRZ], and this leads to explicit generators and relations presenting these subgroups. In Sect. 5 we characterise the commensurator of Bn as a certain stabilizer, under the identi®cation of Bm as the mapping class group of the m-punctured 2-dimensional disk. This identi®cation gives rise to in®nitely many natural ``geometric'' inclusions of Bn in Bm, which we demonstrate to be mutually incommensurable, although they are all conjugate. Indeed, this point of view a€ords a simple description of the centralisers, normalisers and commensurators of all the geometric braid subgroups (Theorem 5.3). In Sect. 6 we show that the action of Bm, as the mapping class group, upon appropriate sets of curves, is a transitive and large action in the sense of Burger and de la Harpe [BH]. The paper concludes with some applications regarding unitary representations of the braid groups, and those induced by braid subgroups. Here is a summary of our principal results regarding the Bn sequence; the relevant de®nitions will follow.