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 $G\to M_{k,0}\to M_{k,\pi }\to M_{k,\pi \mathrm{/}2}$ for $k\in \mathbb{Z}$. 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 $S^1\times {\text{Heis}}_3(\mathbb{R})\mathrm{/}{\mathrm{\Gamma }}_k$ where ${\mathrm{\Gamma }}_k$ 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 $\mathbb{R}\times {\text{Heis}}_3(\mathbb{R})$ but not under the oscillator group $G$.