Presentations of the braid group of the complex reflection group $G(d,d,n)$

TL;DR

This paper explores the structure of the braid group associated with the complex reflection group G(d,d,n), showing its relation to orbifold braid groups and providing compatible presentations based on tagged triangulations.

Contribution

It establishes that the braid group of G(d,d,n) is an index d subgroup of an orbifold braid group and offers new compatible presentations linked to tagged triangulations.

Findings

Braid group of G(d,d,n) is an index d subgroup of the orbifold braid group.

Provides compatible presentations for each tagged triangulation.

Connects presentations to Broué-Malle-Rouquier framework.

Abstract

We show that the braid group associated to the complex reflection group $G(d,d,n)$ is an index $d$ subgroup of the braid group of the orbifold quotient of the complex numbers by a cyclic group of order $d$. We also give a compatible presentation of $G(d,d,n)$ and its braid group for each tagged triangulation of the disk with $n$ marked points on its boundary and an interior marked point (interpreted as a cone point of degree $d$) in such a way that the presentations of Brou\'{e}-Malle-Rouquier correspond to a special tagged triangulation.