On normal subgroups in the fundamental groups of complex surfaces

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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 finitely-presentable, 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 PU(2,1).