Fundamental Group of some Genus-2 Fibrations and Applications

We will prove that given a genus-2 fibration $f:X\to C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to ${\pi }_1(C)\times {\pi }_1(E)$, where $E$ is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces $X$ with genus-2 fibration $X\to C$ such that $b_1(X)>b_1(C)$.