  In this article we introduce the notion of a k-almost-quasifibration and give
many examples. We also show that a large class of these examples are not
quasifibrations. As a consequence, supporting the Asphericity conjecture of
[19], we deduce that the fundamental group of the orbit configuration space of
an effective and properly discontinuous action of a discrete group, on an
aspherical 2-manifold with isolated fixed points is torsion free. Furthermore,
if the 2-manifold has at least one puncture then it is poly-free, and hence has
an iterated semi-direct product of free groups structure, which generalizes a
result of Xicotencatl ([27], Theorem 6.3).
