约化对称空间的Zariski稠密不连续曲面群
Zariski dense discontinuous surface groups for reductive symmetric spaces
AI总结：
本文研究约化型齐性空间与对称空间的不连续曲面群形变，证明高亏格且Zariski闭包局部同构于$SL(2,\mathbb{R})$的标准不连续群存在非标准小形变，并得到对称空间中存在Zariski稠密的高亏格离散曲面子群的结论。
摘要：
设$G/H$是$H$非紧的约化型齐性空间。关于$G/H$的不连续群的形变研究由T. Kobayashi开创。本文证明，若标准不连续群$\Gamma$同构于高亏格曲面群，且其Zariski闭包局部同构于$SL(2,\mathbb{R})$，则$\Gamma$作为$G/H$的不连续群存在非标准的小形变。此外我们还证明，若$G/H$是对称空间且存在非虚阿贝尔不连续群，则$G$中包含一个高亏格的Zariski稠密离散曲面子群，它在$G/H$上真不连续地作用。作为证明的关键部分，我们证明：对于约化群$G$中高亏格的离散曲面子群$\Gamma$，若$\Gamma$的Zariski闭包局部同构于$SL(2,\mathbb{R})$，则$\Gamma$在$G$中存在一个小形变，其Zariski闭包是与$G$实秩相同的约化子群。
英文摘要：
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 $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
