Deformations of Standard Locally Homogeneous Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913950045569024 |
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| author | Kannaka, Kazuki Kobayashi, Toshiyuki |
| author_facet | Kannaka, Kazuki Kobayashi, Toshiyuki |
| contents | Let $X=G/H$ be a homogeneous space, where $G \supset H$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deformations of Standard Locally Homogeneous Spaces Kannaka, Kazuki Kobayashi, Toshiyuki Differential Geometry Primary:57S30, 58H15. Secondary:22D50, 22E40, 22E46, 53C30, 58J50 Let $X=G/H$ be a homogeneous space, where $G \supset H$ 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. |
| title | Deformations of Standard Locally Homogeneous Spaces |
| topic | Differential Geometry Primary:57S30, 58H15. Secondary:22D50, 22E40, 22E46, 53C30, 58J50 |
| url | https://arxiv.org/abs/2507.14832 |