We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$ $\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure braid group.
Moreover, the reflection group that is associated to $A_\ell$ is the symmetric group $S_\ell$, and it is known that there is an exact sequence $PB_\ell \rightarrow B_\ell \rightarrow S_\ell$.
My question is the following: let $L$ be a reflection arrangement (associated to the reflection group $G_L$) in $\mathbb{C}^\ell$. What is the connection between $\pi_1(\mathbb{C}^\ell - L)$ and $G_L$, or the Artin group associated to $G_L$?
Thank you!
I have found out that Brieskorn proved the following (using the above notations): $$ \pi_1(\mathbb{C}^\ell - A_\ell) \cong \text{ker}(A_L \rightarrow G_L) $$ where $A_L$ is the corresponding Artin group, $G_L$ the reflection group.