具有庞特里亚金曲面边界的双曲群的离散嵌入
Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries
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中文总结 AI 辅助
该研究构造了神经与$X_p$同胚的双曲直角Coxeter群，将其离散忠实凸紧嵌入到Isom($\mathbf H^5$)，证明五维最优且$p=2,3$时得循环对称无限族。
中文摘要 AI 辅助
设$X_p$是闭圆盘在旋转$2\pi/p$角下粘合边界点得到的商空间。对每个$2\leq p\leq8$，我们构造一个双曲直角Coxeter群，其神经与$X_p$同胚，该群可容许一个离散、忠实、凸紧的反射表示到Isom($\mathbf H^5$)中，其极限集与指数为$p$的庞特里亚金曲面$\Pi_p$同胚。五维是最优的，因为$\Pi_p$不能嵌入到$\mathbb S^3$中。对$p=2,3$，这些构造是解析的，且产生循环对称的无限族。
英文摘要
Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2π/p$. For every $2\leq p\leq8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Isom$(\mathbf H^5)$, whose limit set is homeomorphic to the index-$p$ Pontryagin surface $Π_p$. Dimension five is optimal, since $Π_p$ does not embed in $\mathbb S^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.
