Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma\_i\in A$. Let $P$ be the "pure braid group", the kernel of the natural homomorphism from $A$ to $W$. Obvious elements in $P$ include the squares $\sigma\_i^2$ of the standard Coxeter generators, as well as conjugates of the $\sigma\_i^2$. Do conjugates of the $\sigma\_i^2$ generate the pure braid group $P$? If so, is there a nice presentation for $P$ in carefully chosen such conjugates?



Generators of pure braid groups of arbitrary Coxeter groups
This note


<http://arxiv.org/pdf/1511.08731v3.pdf>


(unfortunately written in French), Corollaire 3.7, answers your question.


Writing $\bf{W}$ for the canonical positive lift of $W$ in the Artin-Tits group $B\_W$, the pure braid group is generated by the elements of the form $\bf{w}\bf{s}^2 \bf{w}^{-1}$ where $\bf{w}\in\bf{W}$, $\bf{s}$ is the lift in $\bf{W}$ of a simple reflection $s\in S$ and $\bf{w}\bf{s}$ also lies in $\bf{W}$. A presentation in terms of these generators is given in Corollaire 3.7 of the above given link. 


It works for arbitrary (finitely generated) Coxeter groups. 


