  We propose two definitions of configuration Lie groupoids and in both the
cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie
groupoids. We show that this is the best possible extension, in the sense that,
for the class of Lie groupoids corresponding to global quotient orbifolds with
nonempty singular set, the fibration theorems do not hold. Secondly, we prove a
short exact sequence of fundamental groups (called {\it pure orbifold braid
groups}) of one of the configuration Lie groupoids of the Lie groupoid
corresponding to the punctured complex plane with cone points. This shows the
possibility of a quasifibration type Fadell-Neuwirth theorem for Lie groupoids.
As consequences, first we see that the pure orbifold braid groups have
poly-virtually free structure, which generalizes the classical braid group
case. We also provide an explicit set of generators of the pure orbifold braid
groups. Secondly, we prove that a class of affine and finite complex Artin
groups are virtually poly-free, which partially answers the question if all
Artin groups are virtually poly-free ([[4], Question 2]). Finally, combining
this poly-virtually free structure and a recent result ([5]), we deduce the
Farrell-Jones isomorphism conjecture for the above class of orbifold braid
groups. This also implies the conjecture for the case of the affine Artin group
of type $\tilde D_n$, which was left open in [[24], Problem].

Geometric Topology;Algebraic Topology;Group Theory;K-Theory and Homology