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Deformations and rigidity of lattices in solvable Lie groups

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice $Γ$ in $G$ is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of $G$ is connected. This implies that every lattice in a solvable Lie group virtually embeds as a Zariski-dense lattice with finite deformation space. We give examples of solvable Lie groups $G$ which admit Zariski-dense lattices $Γ$ such that $\mathcal{D}(Γ,G)$ is countably infinite, and also examples where the maximal nilpotent normal subgroup of $G$ is connected and simultaneously $G$ has lattices with uncountable deformation space.