Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m\_{ij}$ . The presentation of $W$ is given by 
$$W= < T\_1, \dots, T\_n | T\_i^2=1, T\_iT\_jT\_i \ldots = T\_jT\_iT\_j \ldots, i \neq j>$$ 
where each side of the second equation has $m\_{ij}$ terms. 


Suppose $B\_{W}$ is the corresponding braid group (aka Artin group) obtained by removing the relations $T\_i^2=1$: 
$$B\_{W}=< T\_1, \dots, T\_n | T\_iT\_jT\_i \cdots = T\_jT\_iT\_j \cdots>.$$ 


Then there is a canonical surjective homomorphism $B\_{W} \to W$, the kernel of this homomorphism is called pure Artin group. 


Does anyone know the presentation of the pure Artin group?


Thank you!



Presentation of the pure Artin groups
I don't know any reference where such a presentation is written down for any Artin group. But:


* You can find In this paper of Enriquez a presentation for the pure Artin group of type B: <http://arxiv.org/abs/math/0408035> Proposition 1.1 (by setting $N=2$ is the formulaes)
* For all the infinite families, the corresponding pure braid groups are iterated semi-direct products of free groups. (For all the families but the $D\_n$ one, it follows from the fact that the corresponding hyperplane arrangements are of fiber type, hence these are even almost-direct product). It should leads quite easily to a nice presentation of these groups.


**Edit:** You may also be interested by this paper of Crips and Paris: <http://arxiv.org/abs/math/0210438>. Recall that $W\_{B\_n}=(\mathbb{Z}/2\mathbb{Z})^n \rtimes S\_n$ and that $W\_{D\_n}=(\mathbb{Z}/2\mathbb{Z})^{n-1}\rtimes S\_n$. It is proved in this paper that $B\_{B\_n}=F\_n \rtimes B\_n$ and that $B\_{D\_n}=F\_{n-1}\rtimes B\_n$ where $B\_n$ is the braid group of type $A$ and $F\_n$ is a free group, and that the canonical projection from the braid group to the Coxeter group is compatible with these decomposition (in the sense that in both cases it restricts to the obvious projections $B\_n \rightarrow S\_n$ and $F\_k \rightarrow (\mathbb{Z}/2\mathbb{Z})^k$).


