  In this work we study smooth complex quasi-projective surfaces whose
fundamental group is a free product of cyclic groups. In particular, we prove
the existence of an admissible map from the quasi-projective surface to a
smooth complex quasi-projective curve. Associated with this result, we prove
addition-deletion Lemmas for fibers of the admissible map which describe how
these operations affect the fundamental group of the quasi-projective surface.
Our methods also allow us to produce curves in smooth projective surfaces whose
fundamental groups of their complements are free products of cyclic groups,
generalizing classical results on $C_{p,q}$ curves and torus type projective
sextics, and showing how general this phenomenon is.
