On normal subgroups in the fundamental groups of complex surfaces

Abstract

We show that for each aspherical compact complex surface $X$ whose fundamental group $π$ fits into a short exact sequence $$ 1\to K \to π\to π_1(S) \to 1 $$ 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 $f: X\to S$ 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 $PU(2,1)$.