AI 中文总结
该研究利用酉表示理论，证明临界指数足够接近对称空间体积增长熵的离散子群具有Zariski稠密性，并确定了三类双曲空间等距群下的临界指数最优值，推广了Borel稠密性定理。
AI 中文摘要
设$G$是中心有限且无紧因子的连通实半单线性代数群，$X$为其对应的黎曼对称空间，$h_{\mathrm{vol}}(X)$为$X$的体积增长熵。我们证明存在$\varepsilon=\varepsilon(G)>0$，使得以下结论成立：若$\Gamma<G$是临界指数大于$h_{\mathrm{vol}}(X)-\varepsilon$的离散子群，则$\Gamma$在$G$中是Zariski稠密的。若进一步假设$G$同构于实、复或四元数双曲空间的等距群之一，我们确定了保证对应子群Zariski稠密的临界指数的最小可能值（即$\varepsilon=\varepsilon(G)$的最大可能值）。在无紧因子的实半单李群的离散子群框架下，这一结果推广了关于格的Borel稠密性定理。\n 证明的关键要素来自酉表示理论的成果，特别是Benoist--Liang的近期工作（该工作受Benoist--Kobayashi早期工作的启发），其研究了$G$的任意闭子群$H$对应的$L^2(G/H)$拟正则表示的广义 tempered 性准则。
英文摘要
Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $ε= ε(G) > 0$ so that the following holds: if $Γ< G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - ε$, then $Γ$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $ε= ε(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist--Liang (which was inspired by earlier work of Benoist--Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.
Comments15 pages, no figures. Comments welcome!
