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)
The word "isometries" isn't really appropriate here, because there's no canonical metric on an arbitrary Lie group. But something very close to what you suggested is true: Every Lie group $G$ has a universal covering group that is also a Lie group, unique up to isomorphism; and $G$ is isomorphic to the quotient of its universal covering group by a discrete central subgroup. (One place to find a proof of this is my Introduction to Smooth Manifolds (2nd ed.), Theorem 21.32 and Problem 21.18.)