Fundamental groups of Galois closures of generic projections

For the Galois closure $X_{{\rm gal}}$ of a generic projection from a surface $X$, it is believed that $\pi_1(X_{{\rm gal}})$ 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 $\pi_1(X_{{\rm gal}})$ that depends on $\pi_1(X)$ and data from the generic projection only. In all known examples, this quotient is in fact isomorphic to $\pi_1(X_{{\rm gal}})$. As a byproduct, we simplify the computations of Moishezon, Teicher and others.