We study compact K\"ahler threefolds X with infinite fundamental group whose
universal cover can be compactified. Combining techniques from L2 -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.Comment: 26 pages, no figure. Comments are welcom