It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic to an isometry). On the other hand, closed Euclidean 3-manifolds admit non-trivial Teichmüller spaces of Euclidean structures. What is known about more exotic Thurston's geometries, namely, Nil, $\widetilde{\rm SL}_2(\mathbb R)$ and Sol?

Let $M$ be a closed 3-manifold, $G$ be one of the mentioned Lie groups and $\mathcal T(M, G)$ be the space of the respective geometric structures on $M$ up to isotopy (say, endowed with the topology inherited from the respective space of representations of $\pi_1(M)$). Could it be more than one point? If yes, what is known about it?