Zariski-Dense Deformations of Standard Discontinuous Groups for Pseudo-Riemannian Homogeneous Spaces

Let \(X=G/H\) be a homogeneous space of a Lie group G. When the isotropy subgroup H is non-compact, a discrete subgroup \(\Gamma \) may fail to act properly discontinuously on X. In this article, we address the following question: in the setting where G and H are reductive Lie groups and \(\Gamma \backslash X\) is a standard quotient, to what extent can one deform the discrete subgroup \(\Gamma \) while preserving the proper discontinuity of the action on X? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients \(\Gamma \backslash X\) , when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur.