On normal subgroups in the fundamental groups of complex surfaces

We show that for each aspherical compact complex surface X whose fundamental group π fits into a short exact sequence 1 → K → π → π1(S) → 1 where S is a compact hyperbolic Riemann surface and the group K is finitelypresentable, there is a complex structure on S and a nonsingular holomorphic fibration f : X → S which induces the above short exact sequence. In particular, the fundamental groups of compact complex-hyperbolic surfaces cannot fit into the above short exact sequence. As an application we give the first example of a non-coherent uniform lattice in Isom(H2 C ).