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This paper dates back to 1999 but was never published. The major part of itwas 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 werenot 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 aCoxeter Group W to the semi-direct product of ZT (where T is the set ofreflections) by W. This morphism lifts the map N from W to (Z/2Z)T defined byDyer in [Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57--73]and its restriction to the pure braid group P identifies with theabelianization morphism.  In section 6 we prove that our morphism gives the cocycle associated to theextension B/(P,P) of W by P/(P,P). This extension has been studied recently byBeck [Abelianization of subgroups of reflection groups and their braid groups:an application to cohomology. Manuscripta Math. 136 (2011) 273--293] and byMarin [Reflection groups acting on their hyperplanes J. Algebra 322 (2009)2848--2860] and [Crystallographic groups and flat manifolds from complexreflection groups, arXiv 1509.06213].

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