2-桥链Heckoid轨道折叠中2-桥球上的简单环
Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links
摘要：
在Riley的工作之后，对于每个斜率为$r\in\QQ$的2桥链$K(r)$和大于1的整数或半整数$n$，我们引入{ \it Heckoid orbifold $\orbs(r;n)$ }和{ \it Heckoid群$\Hecke(r;n)=π_1(\orbs(r;n))$，其指数为$K(r)$的$n$}。当$n$为整数时，$\orbs(r;n)$称为{ \it偶} Heckoid orbifold;在这种情况下，底层空间是$K(r)$的外部，奇异集是$K(r)$的下隧道，索引为$n$。本说明的主要目的是宣布以下问题的答案，甚至赫科德轨道。(1)对于$\orbs(r;n)$中由$K(r)$的2-桥球决定的4-穿孔球$\PConway$上的本质简单环，当它在$\orbs(r;n)$中是零同伦的？(2)对于$\PConway$上的两个不同的基本简单环，它们在$\orbs(r;n)$中什么时候是同伦的？我们还宣布应用这些结果的字符品种，McShane的身份，并epimorphisms从2桥链接群到Heckoid群。
英文摘要：
Following Riley's work, for each 2-bridge link $K(r)$ of slope $r\in\QQ$ and an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid orbifold $\orbs(r;n)$} and the {\it Heckoid group $\Hecke(r;n)=π_1(\orbs(r;n))$ of index $n$ for $K(r)$}. When $n$ is an integer, $\orbs(r;n)$ is called an {\it even} Heckoid orbifold; in this case, the underlying space is the exterior of $K(r)$, and the singular set is the lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a 4-punctured sphere $\PConway$ in $\orbs(r;n)$ determined by the 2-bridge sphere of $K(r)$, when is it null-homotopic in $\orbs(r;n)$? (2) For two distinct essential simple loops on $\PConway$, when are they homotopic in $\orbs(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from 2-bridge link groups onto Heckoid groups.
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