Zariski dense discontinuous surface groups for reductive symmetric spaces
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arXiv
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| Format: | Preprint |
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2023
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| author | Kannaka, Kazuki Okuda, Takayuki Tojo, Koichi |
| author_facet | Kannaka, Kazuki Okuda, Takayuki Tojo, Koichi |
| contents | Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $Γ$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $Γ$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $Γ$ of high genus contained in a reductive group $G$, if the Zariski closure of $Γ$ is locally isomorphic to $SL(2,\mathbb{R})$, then $Γ$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08331 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zariski dense discontinuous surface groups for reductive symmetric spaces Kannaka, Kazuki Okuda, Takayuki Tojo, Koichi Differential Geometry Representation Theory Primary 57S30, Secondary 22E40, 22F30, 30F35, 30F60 Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $Γ$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $Γ$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $Γ$ of high genus contained in a reductive group $G$, if the Zariski closure of $Γ$ is locally isomorphic to $SL(2,\mathbb{R})$, then $Γ$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$. |
| title | Zariski dense discontinuous surface groups for reductive symmetric spaces |
| topic | Differential Geometry Representation Theory Primary 57S30, Secondary 22E40, 22F30, 30F35, 30F60 |
| url | https://arxiv.org/abs/2309.08331 |