# Commuting conjugates of finite-order mapping classes

## TL;DR

This paper characterizes when finite-order mapping classes commute in the mapping class group of a surface, explores their conjugates and lifts, and provides methods to realize certain finite abelian groups as isometry groups.

## Contribution

It establishes necessary and sufficient conditions for commuting conjugates of finite-order mapping classes and offers procedures to realize finite abelian groups as isometry groups.

## Key findings

- Finite-order mapping classes with non-spherical orbifolds have conjugates that lift under any finite cover.
- Torsion elements in the centralizer of irreducible finite-order classes have order at most 2.
- Conditions for primitivity of finite-order mapping classes are derived.

## Abstract

Let $\text{Mod}(S_g)$ 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 for two finite-order mapping classes to have commuting conjugates in $\text{Mod}(S_g)$. 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 cover of $S_g$. Furthermore, we show that any 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 $\text{Mod}(S_g)$ as isometry groups.