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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?

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  • $\begingroup$ It's hard to write the details since I have a baby in my other hand, but this is a fact we've used in more than one of our joint papers: any complexification of a real hyperplane arrangement has a presentation by loops around a single hyperplane. $\endgroup$ – Ben Webster Jan 10 '16 at 12:24
  • $\begingroup$ Yes, but for arbitrary Coxeter groups the associated braid group isn't exactly $\pi_1$ of the complexification of a real hyperplane arrangement. I think you're right that the same statement should hold, but I'm actually not sure what the precise presentation should be. Glad to hear you can still type with a baby in one hand :) $\endgroup$ – Tony Licata Jan 11 '16 at 17:22
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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.

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