On normal subgroups in the fundamental groups of complex surfaces

arXivmath/9808085MaRDI QIDQ6500993

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Abstract: We show that for each aspherical compact complex surface $X$ whose fundamental group $pi$ fits into a short exact sequence ${\displaystyle }$ 1 o K o pi o pi_1(S) o 1 ${\displaystyle }$ 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 $f:XoS$ 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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