Geometric realizability of epimorphisms to curve orbifold groups

Given a connected dense Zariski open set of a compact K\"ahler manifold $U$, we address the general problem of the existence of surjective holomorphic maps ${F:U\to C}$ to smooth complex quasi-projective curves from properties of $\pi_1(U)$. It is known that, if such $F$ exists, then there exists a finitely generated normal subgroup $K\trianglelefteq\pi_1(U)$ such that $\pi_1(U)/K$ is isomorphic to a curve orbifold group $G$ (i.e. the orbifold fundamental group of a smooth complex quasi-projective curve endowed with an orbifold structure). In this paper, we address the converse of that statement in the case where the orbifold Euler characteristic of $G$ is negative, finding a (unique) surjective holomorphic map $F:U\to C$ which realizes the quotient $\pi_1(U)\twoheadrightarrow \pi_1(U)/K\cong G$ at the level of (orbifold) fundamental groups. We also prove that our theorem is sharp, meaning that the result does not hold for any curve orbifold group with non-negative orbifold Euler characteristic. Furthermore, we apply our main theorem to address Serre's question of which orbifold fundamental groups of smooth quasi-projective curves can be realized as fundamental groups of complements of curves in $\mathbb{P}^2$.