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

### Abstract

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 $PU(2,1)$

Topics: Mathematics - Geometric Topology, Mathematics - Algebraic Geometry
Year: 1998
OAI identifier: oai:arXiv.org:math/9808085

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