Is every Lie group realized as the quotient of its universal covering group by a discrete group of isometries? Basically, a Lie group analog for the uniformization theorem. It seems reasonable but I'm rather ignorant to the theory of Lie groups and can't seem to find a reference.
Edit: I should have mentioned when the Lie group has a universal cover (i.e. path connected, locally path-connected, and semi-locally path connected)