Let [Formula: see text] denote the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text], and let [Formula: see text] be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on [Formula: see text] that realizes [Formula: see text] as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of [Formula: see text]. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation [Formula: see text].

中文翻译：

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曲面上循环动作的几何实现

令[Formula:see text]表示属[Formula:see text]的封闭可定向曲面[Formula:see text]的映射类群，令[Formula:see text]为有限阶。我们给出了一个归纳过程来在 [Formula: see text] 上构造一个显式双曲结构，将 [Formula: see text] 实现为等距。换句话说，这个过程为[公式：见文本]的循环子群的尼尔森实现问题产生了一个明确的解决方案。此外，我们通过展示某些有限阶映射类如何被视为胖图自同构来给出纯粹的组合观点。作为我们实现的应用，我们确定了某些有限阶映射类的最大归约系统的大小。而且，