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Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by $$W= < T_1, \dots, T_n | T_i^2=1, (T_iT_j)^{m_{ij}}=1, i \neq j>$$

Suppose $B_{W}$ is the corresponding braid group(or called Artin group) obtained by removing the relations $T_i^2=1$, that is, $$B_{W}=< T_1, \dots, T_n | T_iT_jT_i \cdots = T_jT_iT_j \cdots>,$$ where each sides have $m_{ij}$ terms.

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!

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!

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by $$W= < T_1, \dots, T_n | T_i^2=1, (T_iT_j)^{m_{ij}}=1, i \neq j>$$

Suppose $B_{W}$ is the corresponding braid group(or called Artin group) obtained by removing the relations $T_i^2=1$, that is, $$B_{W}=< T_1, \dots, T_n | (T_iT_j)^{m_{ij}}=1, i \neq j>$$

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!

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by $$W= < T_1, \dots, T_n | T_i^2=1, (T_iT_j)^{m_{ij}}=1, i \neq j>$$

Suppose $B_{W}$ is the corresponding braid group(or called Artin group) obtained by removing the relations $T_i^2=1$, that is, $$B_{W}=< T_1, \dots, T_n | T_iT_jT_i \cdots = T_jT_iT_j \cdots>,$$ where each sides have $m_{ij}$ terms.

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!

The presentation of an Artin group (or generalized braid group) is well known: $$B_{W}=< T_1, \dots, T_n | T_iT_jT_i\cdots = T_jT_iT_j \cdots >$$ where each sides have $m_{ij}$terms, $m_{ij}= 2, 3, 4, 6 .$ and $W$ is the Weyl group.

Does anyone know the presentation of the pure Artin group, that is, the fundamental group of $\mathbb{C}^n$ removing the walls of the Weyl chambers?

Thank you!

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by $$W= < T_1, \dots, T_n | T_i^2=1, (T_iT_j)^{m_{ij}}=1, i \neq j>$$

Suppose $B_{W}$ is the corresponding braid group(or called Artin group) obtained by removing the relations $T_i^2=1$, that is, $$B_{W}=< T_1, \dots, T_n | (T_iT_j)^{m_{ij}}=1, i \neq j>$$

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!