Braid groups of imprimitive complex reflection groups

Abstract

We obtain new presentations for the imprimitive complex reflection groups of type (de,e,r) and their braid groups B(de,e,r) for d,r≥2. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group B(de,e,r) is a semidirect product of the braid group of affine type A˜r−1 and an infinite cyclic group. Elements of B(de,e,r) are visualised as geometric braids on r+1 strings whose first string is pure and whose winding number is a multiple of e. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group B(de,e,r) is strongly translation discrete.