Pr\'esentation des groupes de tresses purs et de certaines de leurs extensions
Fran\c{c}ois Digne

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…
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Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic Geometry and Number Theory · Advanced Combinatorial Mathematics
