On the fundamental group of hyperelliptic fibrations.

We prove that for a hyperelliptic fibration on a smooth surface (over the field of complex numbers) with irreducible fibers, the image of the fundamental group of the general fiber in the fundamental group of the surface is finite. As a corollary, we prove the Shafarevich conjecture on the holomorphic convexity of the universal cover, of a surface with a hyperelliptic fibration.