Crystallographic groups and flat manifolds from complex reflection groups

TL;DR

This paper explores the construction of crystallographic and Bieberbach groups from complex reflection groups using generalized braid groups, extending ideas from classical braid group theory.

Contribution

It introduces a method to derive crystallographic and Bieberbach groups as (sub)quotients of generalized braid groups linked to complex reflection groups.

Findings

Constructed new classes of crystallographic groups from complex reflection groups.

Extended classical braid group ideas to more general complex reflection groups.

Provided a framework for understanding flat manifolds associated with complex reflection groups.

Abstract

Following an idea of Gon\c{c}alvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.