论文标题
表面上的MetacyClic动作
Metacyclic actions on surfaces
论文作者
论文摘要
令$ \ mathrm {mod}（s_g）$为封闭的定向表面$ s_g $ g \ geq 2 $的映射类组。在本文中，我们得出了必要和充分的条件，在这些条件下，$ \ mathrm {mod}（s_g）$中的两个扭转元素将具有生成有限的梅西克莱克子组的$ \ mathrm {modrm {modrm {mod}（s_g）$。这为有限循环盖下的周期性映射类的提升性问题提供了完整的解决方案。作为主要结果的应用，我们表明$ 4G $是在$ s_g $上的非切片元环节操作的顺序上可实现的上限，并且通过dicyclic群体的行动实现了这种限制。此外，我们将$ \ mathrm {mod}（s_g）$的双环子组的完整表征达到一定的等价性，我们将称为弱共轭。此外，我们表明，$ \ mathrm {mod}（s_g）$的非切割元环子组中的每个定期映射类都是还原。我们提供了必要和充分的条件，在这些条件下，通过分裂的元环节动作对$ s_g $因素进行非切割的元环节作用。最后，我们提供了$ \ mathrm {mod}（s_ {10}）$和$ \ mathrm {modrm {mod}（s_ {11}）$的有限的非切片元素子组的弱共轭类别的完整分类。
Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\mathrm{Mod}(S_g)$. This yields a complete solution to the problem of liftability of periodic mapping classes under finite cyclic covers. As applications of the main result, we show that $4g$ is a realizable upper bound on the order of a non-split metacyclic action on $S_g$ and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that every periodic mapping class in a non-split metacyclic subgroup of $\mathrm{Mod}(S_g)$ is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on $S_g$ factors via a split metacyclic action. Finally, we provide a complete classification of the weak conjugacy classes of the finite non-split metacyclic subgroups of $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.
