Braid groups of imprimitive complex reflection groups

TL;DR

This paper introduces new, positive, and homogeneous presentations for the braid groups of imprimitive complex reflection groups, revealing their structure and automorphisms, and classifying key elements.

Contribution

It provides novel presentations and diagrams for these braid groups, showing their semidirect product structure and geometric visualization, advancing understanding of their algebraic and geometric properties.

Findings

Presentations are positive and homogeneous.

Braid groups are semidirect products of affine braid groups and cyclic groups.

Classified periodic elements and proved roots are conjugacy-unique.

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 $\widetilde A_{r-1}$ 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.