Fundamental Groups of Galois Closures of Generic Projections

For the Galois closure $\Xgal$ of a generic projection from a surface $X$, it is believed that $\pi_1(\Xgal)$ gives rise to new invariants of $X$. However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of $\pi_1(\Xgal)$ that depends on $\pi_1(X)$ and data from the generic projection only. In all known examples except one, this quotient is in fact isomorphic to $\pi_1(\Xgal)$. As a byproduct, we simplify part of the computations of Moishezon, Teicher and others.