LAUSR

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.