Solvable models for Kodaira surfaces

We consider three families of lattices on the oscillator group $G$, which is an almost nilpotent not completely solvable Lie group, giving rise to coverings $G \to M_{k, 0} \to M_{k, \pi} \to M_{k, \pi/2}$ for $k\in \Z$. We show that the corresponding families of four dimensional solvmanifolds are not pairwise diffeomorphic and we compute their cohomology and minimal models. In particular, each manifold $M_{k, 0}$ is diffeomorphic to a Kodaira--Thurston manifold, i.e. a compact quotient $S^1 \times \Heis_3 (\R) /\Gamma_k$ where $\Gamma_k$ is a lattice of the real three-dimensional Heisenberg group $\Heis_3 (\R)$. We summarize some geometric aspects of those compact spaces. In particular, we note that any $M_{k, 0}$ provides an example of a solvmanifold whose cohomology does not depend on the Lie algebra only and which admits many symplectic structures that are invariant by the group $\R \times\Heis_3 (\R)$ but not under the oscillator group $G$.