Pr\'esentation des groupes de tresses purs et de certaines de leurs extensions

TL;DR

This paper explores the structure of pure braid groups associated with Coxeter groups, defining a key morphism that relates to their extensions and abelianization, with implications for understanding their algebraic properties.

Contribution

It introduces a new morphism from braid groups to a semi-direct product, linking it to known reflection subgroup structures and extensions of Coxeter groups.

Findings

Defined a morphism from braid groups to a semi-direct product involving reflections.

Identified the morphism with the abelianization of the pure braid group.

Proved the morphism corresponds to a specific group extension related to Coxeter groups.

Abstract

This paper dates back to 1999 but was never published. The major part of it was included in the joint paper [Digne-Gomi, Presentation of pure braid groups, J. Knot Theory and its Ramifications 10 (2001) 609--623]. Sections 2 and 6 were not included there. They are independent from the rest of the paper. In section 2 we define a morphism from a braid group B associated to a Coxeter Group W to the semi-direct product of ZT (where T is the set of reflections) by W. This morphism lifts the map N from W to (Z/2Z)T defined by Dyer in [Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57--73] and its restriction to the pure braid group P identifies with the abelianization morphism. In section 6 we prove that our morphism gives the cocycle associated to the extension B/(P,P) of W by P/(P,P). This extension has been studied recently by Beck [Abelianization of subgroups of reflection groups and their braid groups: an application to cohomology. Manuscripta Math. 136 (2011) 273--293] and by Marin [Reflection groups acting on their hyperplanes J. Algebra 322 (2009) 2848--2860] and [Crystallographic groups and flat manifolds from complex reflection groups, arXiv 1509.06213].