Orbifold braid groups and complex braid groups

A result of Allock 1 states that certain orbifold braid groups contain Artin groups of type $D_n$, ${\tilde{B}}_n$ and ${\tilde{D}}_n$ as finite index subgroups. The underlying orbifolds have at most two cone points of order two. Based on 10 and 12, we generalize this result allowing cone points of arbitrary order. In these cases, the orbifold braid groups contain similar subgroups of finite index. We show that in many cases these subgroups can be identified as certain complex braid groups.