  We consider the class of quasiprojective varieties admitting a dominant
morphism onto a curve with negative Euler characteristic. The existence of such
a morphism is a property of the fundamental group. We show that for a variety
in this class the number of maps onto a hyperbolic curve or surfaces can be
estimated in terms of the numerical invariants of the fundamental group. We use
this estimates to find the number of biholomorphic automorphisms of complements
to some arrangements of lines.
