Threefolds with quasi-projective universal cover

We study compact K\"ahler threefolds X with infinite fundamental group whose universal cover can be compactified. Combining techniques from $L^2$ -theory, Campana's geometric orbifolds and the minimal model program we show that this condition imposes strong restrictions on the geometry of X. In particular we prove that if a projective threefold with infinite fundamental group has a quasi-projective universal cover, the latter is then isomorphic to the product of an affine space with a simply connected manifold.