Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Let ${\mathbb{R}}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of ${\mathbb{R}}^{2,2}$ is ${\mathbb{R}}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R⋉G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal’bo.