Let $$\text {Mod}(S_g)$$ Mod ( S g ) be the mapping class group of the closed orientable surface $$S_g$$ S g of genus $$g\ge 2$$ g ≥ 2 . In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in $$\text {Mod}(S_g)$$ Mod ( S g ) . As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cyclic cover of $$S_g$$ S g . Furthermore, we show that any nontrivial torsion element in the centralizer of an irreducible finite order mapping class is of order at most 2. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of $$\text {Mod}(S_g)$$ Mod ( S g ) as isometry groups.

中文翻译：

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有限阶映射类的通勤共轭

令$$\text {Mod}(S_g)$$Mod ( S g ) 为属$$g\ge 2$$g ≥ 2 的封闭可定向曲面$$S_g$$ S g 的映射类群。在本文中，我们推导出两个有限阶映射类在 $$\text {Mod}(S_g)$$Mod ( S g ) 中具有交换共轭的充分必要条件。作为这个结果的应用，我们证明了任何有限阶映射类，其对应的轨道不是球体，具有在 $$S_g$$S g 的任何有限片循环覆盖下提升的共轭。此外，我们证明了不可约有限阶映射类的中心器中的任何非平凡扭转元素的阶至多为 2。我们还获得了有限阶映射类的原始性条件。最后，