Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions.


Thanks to Milnor, we know that the $\pi\_1$ any compact manifold with nonnegative Ricci curvature has polynomial growth. 


I want to know if anything is known about the opposite direction : which manifolds whose $\pi\_1$ has polynomial growth admit metric with nonnegative Ricci curvature ?


At least if you allow non compact manifolds, the answer cannot be 'all of them' (because of Whitehead's three manifold), and I feel like there are compact counter examples but I am not able to cook one. 



Converse to Milnor's theorem on manifolds with nonnegative Ricci curvature
For a compact counterexample, take any nilmanifold $N/H$ modulo the action of a freely acting cocompact lattice $\Lambda$, assuming $N/H$ is not just Euclidean space and $\Lambda$ is not just virtually abelian. For instance, $N=N/H$ is the $3 \times 3$ real Heisenberg group and $\Lambda$ is the $3 \times 3$ integer Heisenberg group. The proof that this is a counterexample is to apply Wilking's theorem in Anton's answer, and the theorem that a finitely generated nilpotent group has polynomial growth.


