Hyperbolic geometry and fundamental groups of complex 3-folds

We give an alternative short proof of Taubes' theorem stating that compact complex 3-folds can have arbitrary finitely presented fundamental group. This is related to a question of Gromov: Is it true that every compact manifold is homeomorphic to a quotient of the hyperbolic space $H^n$ by an isometric (non-free) action of a discreet group? This talk is based on a joint work with Anton Petrunin.