On normal subgroups in the fundamental groups of complex surfaces

Details On normal subgroups in the fundamental groups of complex surfaces On normal subgroups in the fundamental groups of complex surfaces

1998-08-18

arxiv.org/abs/math/9808085v1

Mathematics

Geometric Topology

Scientific paper

We show that for each aspherical compact complex surface $X$ whose fundamental group $\pi$ fits into a short exact sequence $$ 1\to K \to \pi \to \pi_1(S) \to 1 $$ 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 $PU(2,1)$.

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