Deformations of Standard Locally Homogeneous Spaces

TL;DR

This paper investigates how discrete subgroups acting on homogeneous spaces can be deformed while maintaining proper discontinuity, providing classification results and criteria for rigidity and deformations.

Contribution

It offers new classification results and criteria for deformations of standard quotients of homogeneous spaces by discrete groups.

Findings

Conditions for local rigidity of compact quotients

Criteria for deforming standard to nonstandard quotients

Characterization of Zariski-closure of groups under deformation

Abstract

Let $X=G/H$ be a homogeneous space, where $G \supset H$ are reductive Lie groups. We ask: in the setting where $\Gamma \backslash G/H$ is a standard quotient, to what extent can the discrete subgroup $\Gamma$ be deformed while preserving the proper discontinuity of the $\Gamma$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $\Gamma\backslash X$; 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.