On normal subgroups in the fundamental groups of complex surfaces

TL;DR

This paper investigates the structure of fundamental groups of certain complex surfaces, establishing conditions for holomorphic fibrations and providing examples of non-coherent lattices in complex hyperbolic geometry.

Contribution

It characterizes when the fundamental group of an aspherical complex surface admits a specific short exact sequence, linking algebraic and geometric structures.

Findings

Existence of holomorphic fibrations inducing the exact sequence

Fundamental groups of complex-hyperbolic surfaces do not fit the sequence

First example of a non-coherent uniform lattice in PU(2,1)

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 $PU(2,1)$.