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In this article we introduce the notion of a k-almost-quasifibration and givemany examples. We also show that a large class of these examples are notquasifibrations. As a consequence, supporting the Asphericity conjecture of[19], we deduce that the fundamental group of the orbit configuration space ofan effective and properly discontinuous action of a discrete group, on anaspherical 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 hasan iterated semi-direct product of free groups structure, which generalizes aresult of Xicotencatl ([27], Theorem 6.3).

Almost-quasifibrations and fundamental groups of orbit configuration spaces