There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. He showed that solvable Lie groups have the same homology with finite coefficients as their underlying discrete groups. Morel recently announced a proof of the Friedlander-Milnor conjecture that implies this is also true for complex algebraic Lie groups.
I've always been curious about this problem, and I've discussed it with a number of other mathematicians. One question that I haven't been able to sort out about it was to what extent the homology of a Lie group with coefficients in an arbitrary abelian group $A$ made discrete is determined, e.g. whether the homology groups being nonzero, or finitely generated, is independent of ZFC. This problem involves aspects of both manifolds and the projective resolution of modules, and hence it might be plausible that this is the case.
Are there any results, positive or negative, about the dependence of (co)homology of $G^\delta$ on the underlying model of set theory?