论文标题
在表面上拆分元环节动作
Split 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 $的映射类组。在本文中，我们得出了必要和充分的条件，在这些条件下，在$ \ mathrm {mod}（s_g）$中的两个扭转元素将具有生成有限的分式分配的非亚伯利亚梅西克莱克子组的共轭物的$ \ m m astarrm {modrm {mod}（mod}（s_g）$。作为主要结果的应用，我们将有限的二面体和$ \ mathrm {mod}（s_g）$的广义Quaternion子组的完整表征达到一定的等价性，我们称之为弱共轭。此外，我们表明，任何有限订单映射类，其相应的Orbifold是一个球体，具有共轭物，可在某些有限的定期循环盖（$ s_g $）下举起。此外，对于$ g \ geq 5 $，我们展示了由$ \ mathrm {mod}（s_g）$的无限二面亚组的存在，该子组是由涉及的互动和一个界对$ 3 $的界对生成的。最后，我们提供了$ \ mathrm {mod}（s_3）$和$ \ mathrm {modrm {mod}（s_5）$的非亚伯有限拆分子组的弱共轭类（s_5）$的完整分类。我们还描述了其中一些动作的非平凡的几何实现。
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 split non-abelian metacyclic subgroup of $\mathrm{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of $S_g$. Moreover, for $g \geq 5$, we show the existence of an infinite dihedral subgroup of $\mathrm{Mod}(S_g)$ that is generated by an involution and a root of a bounding pair map of degree $3$. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of $\mathrm{Mod}(S_3)$ and $\mathrm{Mod}(S_5)$. We also describe nontrivial geometric realizations of some of these actions.
