Fundamental group of some genus-$2$ fibrations and applications

We will prove that given a genus-$2$ fibration $f\colon X \to C$ 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 $b_1(X)>b_1(C)$. This is joint work with Sagar Kolte.