Deformations of Standard Locally Homogeneous Spaces

Let $X=G\mathrm{/}H$ be a homogeneous space, where $G\supset H$ are reductive Lie groups. We ask: in the setting where $\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H$ is a standard quotient, to what extent can the discrete subgroup $\mathrm{\Gamma }$ be deformed while preserving the proper discontinuity of the $\mathrm{\Gamma }$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $\mathrm{\Gamma }\mathrm{\backslash }X$; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur. Proofs of the results stated in this paper are provided in detail in arXiv:2507.03476.