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 $d,r\ge 2$. 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 ${\tilde{A}}_{r-1}$ and an infinite cyclic group. Elements of $B(de,e,r)$ are visualized 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.

published version

References in corpus (8)

• Non-positively curved aspects of Artin groups of finite type
• The n-th root of a braid is unique up to conjugacy
• Garside groups are strongly translation discrete
• A new Garside structure for braid groups of type $(e,e,r)$
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Non-crossing partitions of type (e,e,r)
• Periodic elements in Garside groups
• Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

Cited by in corpus (6)

• Configuration Lie groupoids and orbifold braid groups
• Interval Structures, Hecke Algebras, and Krammer's Representations for the Complex Braid Groups B(e,e,n)
• Some virtually poly-free Artin groups
• On torus knot groups and a submonoid of the braid group
• Interval Garside Structures Related to the Affine Artin Groups of Type $\tilde{A}$
• The isomorphism conjecture for Artin groups