Deformations of Standard Locally Homogeneous Spaces

2025/07/20 by Kannaka, Kazuki, Kobayashi, Toshiyuki
#22E40 #22E46 #53C30 #58H15. Secondary:22D50 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Primary:57S30

paper · doi:10.48550/arxiv.2507.14832

Abstract

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

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