论文标题
映射类组的无限元环节子组
Infinite metacyclic subgroups of the mapping class group
论文作者
论文摘要
对于$ g \ geq 2 $，令$ \ text {mod}（s_g）$是封闭的可定向表面$ s_g $ g $ g $的映射类组。在本文中，我们为存在$ \ text {mod}（s_g）$的无限metacyclic子组提供了必要的条件。特别是，我们提供了必要和充分的条件，在这些条件下，伪anosov映射类生成了$ \ text {mod}（s_g）$的无限metacyclic子组，并使用非平凡的周期性映射类。作为我们主要结果的应用，我们确定了$ \ text {mod}（s_g）$ isomorphic to $ \ mathbb {z} \ rtimes \ mathbb {z} _m， $ \ mathbb {z} \ rtimes \ mathbb {z} $。此外，我们按照实现的$ \ text {mod}（s_g）$的无限元元素子组的非平地周期发电机的顺序得出界限。最后，我们表明，不可约定的周期性映射类$ f $的中心设备是$ \ langle f \ rangle $或$ \ langle f \ rangle \ times \ times \ langle i \ rangle $，其中$ i $是过度易ellirtiriptic的参与。
For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and sufficient conditions for the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of $\text{Mod}(S_g)$ with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$ isomorphic to $\mathbb{Z}\rtimes \mathbb{Z}_m, \mathbb{Z}_n \rtimes \mathbb{Z}$, and $\mathbb{Z} \rtimes \mathbb{Z}$. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of $\text{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class $F$ is either $\langle F\rangle$ or $\langle F\rangle \times \langle i\rangle$, where $i$ is a hyperelliptic involution.
