On the fundamental group of hyperelliptic fibrations and some applications

We prove that for a hyperelliptic fibration on a surface of general type with irreducible fibers over a (possibly) non-complete curve, the image of the fundamental group of a general fiber in the fundamental group of the surface is finite. Examples show that the result is optimal. As a corollary of this result we prove two conjectures; the Shafarevich conjecture on holomorphic convexity for the universal cover of these surfaces, and a conjecture of Nori on the finiteness of the fundamental groups of some surfaces. We also prove a striking general result about the multiplicities of multiple fibers of a hyperelliptic fibration on a smooth, projective surface.