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?
Best Answer
$\newcommand{\PSLt}{\widetilde{\mathrm{PSL}_2}}$Manifolds with $\PSLt$ geometry can have non-trivial moduli. For example, suppose that $S$ is a closed, connected, oriented surface with genus at least two. Let $M$ be a copy of $\mathrm{UT}S$, the unit tangent bundle of $S$. Then there is a (distinct) $\PSLt$ structure on $M$ coming from every (distinct) hyperbolic structure on $S$.
The argument is similar when $M$ is instead the unit tangent bundle over a hyperbolic orbifold $B$. The exception is when $B$ is a triangle orbifold: that is, a copy of the two-sphere with three cone points having angles $2\pi/p$, $2\pi/q$, and $2\pi/r$ so that $1/p + 1/q + 1/r > 1$. In this case $B$ has a unique hyperbolic structure (the double of the corresponding triangle); also $\mathrm{UT}B$ has a unique $\PSLt$ structure.
The story for $\mathbb{H}^2 \times \mathbb{R}$ geometry follows similar lines.