A GEOMETRIC AND ALGEBRAIC DESCRIPTION OF ANNULAR BRAID GROUPS

¢¡¤£¦¥¨§¤©¨��¥� � We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn−1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph Ãn−1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear. In this paper we examine a variation on the classical braid groups known as the annular braid groups. We denote by CBn the annular braid group on n strings. A precise definition is given below. In section two we will examine the geometry of CBn and show that CBn is isomorphic to Dn+1. The group Dn+1, first studied b