

\section{automorphism of the Riemann sphere}

    In this section, we will obtain the Lemma \ref{} which is often used in this paper.
    
    Since any sphere is conformally equivalent to the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C}\ ^{\cup} \{ \infty \}$, we will consider the automorphism of $\hat{\mathbb{C}}$.
    
    Firstly, we have the following fact.
    
        \begin{prop}
            For arbitrary automorphism $f$ of $\hat{\mathbb{C}}$, there exist complex numbers $a,b,c,d$ such that $ad-bc \neq 0$ and $f(Z) = \frac{az+b}{cz+d}$.
        \end{prop}
        
        \begin{prop}
            Take arbitrary mutually distinct points $z_1,z_2,z_3$ of $\hat{\mathbb{C}}$.
            For mutually distinct points $w_1,w_2,w_3$, there uniquely exists the automorphism $f$ such that $f(z_i) = w_i$.
        \end{prop}
        
        \begin{lem}
            If an automorphism of $S^2$ fixes mutually distinct three points, then the automorphism is $\operatorname{id}_{S^2}$.
        \end{lem}
        
        \begin{proof}
            Take an automorphism $f$ which fixes mutually distinct points $z_1, z_2, z_3$.
            By Proposition \ref{}, there uniquely exists the automorphism $g$ such that, $g(z_1)=0, g(z_2)=1$ and $g(z_3)= \infty$.
            Let us consider the automorphism $h = g\circ f \circ g^{-1}$.
            By Proposition \ref{}, there exists complex numbers $a,b,c,d$ such that $ad-bc \neq 0$ and $h(z) = \frac{az+b}{cz+d}$.
            Since $h$ fixes $0,1$ and $\infty$, we have $\frac{b}{d} = 0, \frac{a+b}{c+d} = 1$ and $\frac{a}{c} = \infty$.
            This implies that $b=c=0$ and $a=d$, and hence, $h= \operatorname{id}_{\hat{\mathbb{C}}}$.
            Thus, we have $f = g^{-1} h g = \operatorname{id}_{\hat{\mathbb{C}}}$.
            This completes the proof of Lemma \ref{}.
        \end{proof}



\section{Periodic diffeomorphisms of tori}

	We classify actions of finite abelian groups on $T^{2}$ by using the well-known classification of wall paper groups.
	Let $G$ be a finite abelian subgroup of $\operatorname{Diff}_+(T^2)$.
	Since $\mathbb{C}$ is the universal covering space of $T^2$ and $G$ preserves the orientation of $T^2$, we can lift $G$ to one of the wallpaper groups $p1, p2, p3, p4$ and $p6$ as shown in Figures \ref{fig:p1}, \ref{fig:p2}, \ref{fig:p3}, \ref{fig:p4} and \ref{fig:p6}, respectively.
		
		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p1.pdf}
				\caption{}
				\label{fig:p1}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p2.pdf}
				\caption{}
				\label{fig:p2}
			\end{minipage}\\
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p3.pdf}
				\caption{}
				\label{fig:p3}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p4.pdf}
				\caption{}
				\label{fig:p4}
			\end{minipage}\\
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p6.pdf}
				\caption{}
				\label{fig:p6}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
			\end{minipage}
		\end{tabular}
		\end{figure}
	
	By projecting their group action to $T^2$, the following proposition holds.
	
		\begin{prop}
		\label{prop:torus}
			$G$ is conjugate to a subgroup of a group generated by one of the following generators.
				\begin{enumerate}
					\item $2$ translations as shown in Figure \ref{fig:pp1},
					\item $h_{4,1}^2$ as shown in Figure \ref{fig:pp2},
					\item $h_{3,1}$ and a translation as shown in Figure \ref{fig:111J},
					\item $h_{4,1}$ and a hyperelliptic involution $I$ as shown in Figure \ref{fig:112I},
					\item $h_{6,3}$ as shown in Figure \ref{fig:pp6}.
				\end{enumerate}
		\end{prop}
		
		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{pp2.pdf}
				\caption{}
				\label{fig:pp2}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{123.pdf}
				\caption{}
				\label{fig:pp6}
			\end{minipage}
		\end{tabular}
		\end{figure}
		
	Since $\iota_1$ is a translation on $T^2$, the following holds. 
		
		\begin{cor}
		\label{cor:torus}
		    Any finite abelian subgroup of $\operatorname{Diff}_+(\Sigma_g)$ which contains $\iota_1$ is conjugate to a subgroup of Figures \ref{fig:pp1}, \ref{fig:112I} or \ref{fig:111J}.
		\end{cor}
\section{Introduction}
    
    An involution on an oriented closed surface $\Sigma_{g}$ of genus $g>1$ is called \emph{hyperelliptic} if it fixes $2g+2$ points.
    It is well known that such involution is unique up to conjugacy, and has the maximum number of fixed points among involutions on $\Sigma_{g}$.
    A diffeomorphism of $\Sigma_{g}$ which commutes with a hyperelliptic involution is called \emph{hyperelliptic}.
    Ishizaka \cite{Ishizaka3,Ishizaka} classified hyperelliptic periodic diffeomorphisms up to conjugacy and gave the right-handed Dehn twist presentations of their mapping classes based on resolutions of singularities of families of Riemann surfaces.
    An involution $\iota$ of $\Sigma_{g}$ is hyperelliptic if and only if the quotient space $\Sigma_{g}/\langle\iota\rangle$ is homeomorphic to a sphere.
    In this paper, we will classify periodic diffeomorphisms of $\Sigma_{g}$ which commute with an involution $\iota_{g}$ whose quotient space $\Sigma_{g}/\langle \iota_{g} \rangle$ is homeomorphic to a torus.
    In this case, there is a significant difference in the classification from the hyperelliptic case considered in \cite{Ishizaka}.
    Recall that a diffeomorphism of $\Sigma_g$ is called {\it reducible} if there exists a family of isotopy classes of essential simple closed curves on $\Sigma_g$ preserved by the diffeomorphism and {\it irreducible} if otherwise.
    By the classification in \cite{Ishizaka}, every hyperelliptic periodic diffeomorphism of $\Sigma_{g}$ is a power of an irreducible periodic diffeomorphism.
    In turn, as we will see below, this is not the case for periodic diffeomorphisms which commute with an involution $\iota_{g}$ such that $\Sigma_{g}/\langle \iota_{g} \rangle \cong T^2$.
    (see Example \ref{ex:red}).
    In this article, we focus on the classification in the irreducible case.
    Irreducible periodic diffeomorphisms of $\Sigma_g$ have been studied from various viewpoints (see e.g. \cite{Hirose2,Hirose-Kasahara,Ge-Ra,Dhanwani}).

	To state the result, let us present an example of an irreducible periodic diffeomorphism $h_{n, p}$ of an oriented closed surface parametrized by a positive integer $n \geq 3$ and $p \in \{ 1,\ldots, n-1 \}$.
	Consider a regular $2n$-gon.
	Let $\alpha_i$ and $\beta_i\  ( i = 0, \ldots, n-1 )$ be the edges of the $2n$-gon shown in Figure \ref{fig:Sig}.
	By identifying $\alpha_i$ with $\beta_j$ for every pair of $i$ and $j \in \{ 0, \ldots, n-1 \}$ with $i-j \equiv p \mod n$, we obtain a surface $\Sigma$ of genus $g = \frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}$.
	We denote $h_{n,p}$ the diffeomorphism of $\Sigma$ induced from the clockwise $\frac{2\pi}{n}$-rotation of the $2n$-gon. To simplify the figures, both $\alpha_{0}$ and $\beta_{p}$ will be denoted by $\alpha$, and omit the notation of other edges in the rest of the paper.

		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				 
				  	\includegraphics[width=0.9\textwidth]{Figures/Sig.pdf}
				\caption{A diffeomorphism $h_{n,p}$}
				\label{fig:Sig}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\captionsetup{width=.8\linewidth}
				\includegraphics[width=0.9\textwidth]{Figures/connsum.pdf}
				\caption{Connected sum of some irreducible periodic diffeomorphisms}
				\label{fig:connsum}
			\end{minipage}
		\end{tabular}
		\end{figure}

        \begin{ex}
        \label{ex:red}
            We can construct an example of a reducible periodic diffeomorphism which commutes with an involution $\iota_{2g+1}$ from $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$ by a connected sum (Figure \ref{fig:connsum}):
            Remove four small disks centered at the fixed points of $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$, respectively, and glue the complements of the disks along the circle boundaries so that $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$ coincide with each other on the boundary.
            Then we get a periodic diffeomorphism $f$ on $(\Sigma_g - disks) \cup_{\partial} (\Sigma_g - disks)$ of period $4g$.
            Note that $f^{2g}$ is an involution whose quotient space is a torus which commutes with $f$.
        \end{ex}
    
    Let us state the main result of this article.
    In this result, we classify irreducible periodic diffeomorphisms which commutes with $\iota_g$. 
	
		\begin{thm}
		\label{thm:irr1}
			Let $\iota_g$ be an involution of $\Sigma_g$ such that $\Sigma_g/ \langle \iota_g \rangle$ is homeomorphic to a torus as shown in Figure \ref{fig:iotag}.
			For any periodic diffeomorphism $f$ of $\Sigma_{g}$ which commutes with an involution $\iota_g$, if the subgroup $G=\langle f, \iota_g \rangle$ of $\operatorname{Diff}_{+}(\Sigma_{g})$ is irreducible, then $G$ is conjugate to a subgroup of one of following:
			    \begin{enumerate}
			        \item $\langle h_{6,1}, I \rangle$ (Figure \ref{fig:114}),
			        \item $\langle h_{8,1} \rangle$ (Figure \ref{fig:116}),
			        \item $\langle h_{8,5} \rangle$ (Figure \ref{fig:125}),
			        \item $\langle h_{12,2} \rangle$ (Figure \ref{fig:129}),
			        \item $\langle h_{12,3} \rangle$ (Figure \ref{fig:138}).
			    \end{enumerate}
		\end{thm}
		
			It is remarkable that there are only finite cases in the classification of Theorem \ref{thm:irr1}.
			It is in contrast to the case of the classification of hyperelliptic periodic classes, where we have three infinite families of conjugacy classes. 
	
	
        \begin{figure}[ht]
	    	\centering
		    \includegraphics[scale=0.45]{Figures/iotag.pdf}
    		\caption{The involution $\iota_{g}$}
	    	\label{fig:iotag}
        \end{figure}
        
		\begin{figure}[ht]
		\begin{tabular}{ccc}
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/114.pdf}
				\caption{$h_{6,1}$}
				\label{fig:114}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/116.pdf}
				\caption{$h_{8,1}$}
				\label{fig:116}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/125.pdf}
				\caption{$h_{8,5}$}
				\label{fig:125}
			\end{minipage}\\
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/129.pdf}
				\caption{$h_{12,2}$}
				\label{fig:129}
			\end{minipage} \hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/138.pdf}
				\caption{$h_{12,3}$}
				\label{fig:138}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\captionsetup{width=\linewidth}
				\includegraphics[width=1\textwidth]{Figures/Iphi3.pdf}
				\caption{$h_{2g+2,1}$ and $I$}
				\label{fig:Iphi3}
			\end{minipage}
		\end{tabular}
		\end{figure}

	
	For hyperelliptic involution $I$ and hyperelliptic periodic diffeomorphism $f$, the group $G = \langle f, I \rangle$ is cyclic except the case where $f$ is a power of $h_{2g+2,1}$ (see \cite{TN, Ishizaka}).
	Motivated by this fact, we give a classification of the centralizer of irreducible diffeomorphisms as follows.

        \begin{thm}
		\label{thm:irr2}
    		Let $f$ be an irreducible periodic diffeomorphism of $\Sigma_{g}$ with $g>1$, and $H$ a finite subgroup of the centralizer of $f$ in $\operatorname{Diff}_{+}(\Sigma_{g})$.
    		Then one of the following occurs:
    		
                \begin{enumerate}
                    \item $H$ is a subgroup of $\langle f \rangle$.
                    \item $H$ is conjugate to a subgroup of $\langle h_{4g+2,2g+1} \rangle$ so that $\langle f \rangle$ is conjugate to $\langle h_{2g+1,1} \rangle$.\label{case:irr22}
                    \item $H$ is conjugate to a subgroup of $\langle h_{2g+2, 1}, I \rangle$ so that $\langle f \rangle$ is conjugate to $\langle h_{2g+2,1} \rangle$.
                \end{enumerate}
	   	\end{thm}
		
	As a corollary, we obtain the following characterization of $h_{n,1}$.
		
		\begin{cor}
    		If an irreducible periodic diffeomorphism $f$ on $\Sigma_{g}$ commutes with two involutions $I_1$ and $I_2$ which are not mutually conjugate, then $f$ is conjugate to a power of either $h_{2g+1,1}$ or $h_{2g+2,1}$.   
		\end{cor}


\section{periodic diffeomorphisms and their total valencies}

    
    We first recall the classification of the conjugacy classes of periodic diffeomorphisms on $\Sigma_{g}$ in terms of the total valency introduced in \cite{Ashikaga-Ishizaka}.
	Let $f \in \operatorname{Diff}_+(\Sigma_{g})$ be a periodic diffeomorphism of order $n$ and $C$ the cyclic subgroup of $\operatorname{Diff}_{+}(\Sigma_{g})$ generated by $f$.
	An $C$-orbit $Cx$ is called {\it multiple} if $|Cx| < |C|$, where $|\cdot|$ denote the cardinality.
	For $x \in \Sigma_g$, the minimum positive integer $\ell$ such that $f^{\ell}(x)=x$ is called the period of $Cx$.
	Let $\lambda = |C_{x}|$, where $C_{x}$ is the isotropy group of $x$.
	Note that $\frac{n}{\lambda}$ is equal to the period of $Cx$.
	Then, there uniquely exists $\mu \in \{1,2, \ldots, \lambda - 1\}$ such that that the restriction of $f$ to a small neighborhood of $x$ is the clockwise $\frac{2\pi\mu}{\lambda}$-rotation.
	Since $\mu$ and $\lambda$ are coprime, there exists an integer $\theta \in \{1,2, \ldots, \lambda - 1\}$ such that $\mu \theta \equiv 1 \ \mod \lambda$.
	The {\it valency} of $Cx$ is defined by $\frac{\theta}{\lambda}$.
	Let $\frac{\theta_1}{\lambda_1}, \frac{\theta_2}{\lambda_2}, \ldots, \frac{\theta_s}{\lambda_s}$ be the valencies of all multiple orbits of $f$.
	The data $[ \, g,n \,; \frac{\theta_1}{\lambda_1} + \frac{\theta_2}{\lambda_2} + \cdots + \frac{\theta_s}{\lambda_s} \, ]$ is called the {\it total valency} of $f$.
		
		\begin{figure}[ht]
			\centering
			\includegraphics[width=0.5\textwidth]{Figures/Sig2.pdf}
			\caption{A loop around a vertex $y$}
			\label{fig:Sig2}
		\end{figure}
	
	By the following theorem, total valencies exactly determine periodic diffeomorphisms on $\Sigma_{g}$ up to conjugation.
	
		\begin{thm}[{Nielsen \cite[Section 11]{Nielsen}, see also \cite[Section 1.3]{Ashikaga-Ishizaka}, \cite[Theorem 2.1]{Hirose}}]
		\label{thm:Nielsen}
			Let $f, f'$ be periodic diffeomorphisms of $\Sigma_{g}$ with total valencies $[\,g,n \,;\, \frac{\theta_1}{\lambda_1} + \frac{\theta_2}{\lambda_2} + \cdots +\frac{\theta_s}{\lambda_s} \,]$ and $[\,g,n' \,;\,\frac{\theta_1'}{\lambda_1'} + \frac{\theta_2'}{\lambda_2'} + \cdots +\frac{\theta_{s'}'}{\lambda_{s'}'} \,]$, respectively.
			$f$ is conjugate to $f'$ if and only if the following are satisfied:
				\begin{enumerate}
				\renewcommand{\labelenumi}{(\roman{enumi})}
					\item $s=s'$,
					\item $n=n'$,
					\item after changing indices, we have $\frac{\theta_i}{\lambda_i}=\frac{\theta_i'}{\lambda_i'}$ for $i =1,2,\ldots,s$.
				\end{enumerate}
		\end{thm}
	
	By Theorem \ref{thm:Nielsen}, we can identify the total valency of a periodic diffeomorphism with its conjugacy class.
	We will use the following well-known observations of Nielsen.

		\begin{prop}[{Nielsen \cite[Equation(4.6)]{Nielsen}}]
		\label{prop:Nielsenint}
			The sum of valencies of all multiple orbits of a periodic diffeomorphism is an integer.
		\end{prop}
		
	    \begin{rem}
		\label{rem:inv}
        	Any multiple orbit of an involution is a fixed point whose valency is $\frac{1}{2}$.
			Thus, by Proposition \ref{prop:Nielsenint}, the number of fixed points of an involution is even.
	    \end{rem}

	In general, the composite of two periodic diffeomorphism may not be periodic, and hence, the product of total valency does not make sense.
	But one can define powers of total valency of a periodic diffeomorphism $f$ by the total valency of powers of $f$.

    Let us recall the well-known Riemann-Hurwitz formula for the convenience of the reader.
		
		\begin{prop}[{Riemann-Hurwitz formula}]
		\label{prop:RH}
			Consider the $n$-fold branched covering from $\Sigma_g$ to $\Sigma_{g'}$ with branched indices $\lambda_1, \ldots, \lambda_s$.
			Then we have the formula
				\[
					2g-2 = n \left( 2g' -2 + \displaystyle\sum_{1 \leq i \leq s} \left( 1 - \frac{1}{\lambda_i} \right) \right).
				\]
		\end{prop}

    We will use the following result due to Harvey, which gives a necessary condition for branched indices.
	
		\begin{prop}[{Harvey \cite[Theorem 4]{Harvey}}]
		\label{prop:Harvey}
		    Assume $g>1$.
			Set $M = \operatorname{lcm}(\lambda_1,\lambda_2,\cdots , \lambda_s)$.
				If there is cyclic covering from $\Sigma_g$ to $\Sigma_{g'}$ with branched indices $\lambda_1,\cdots, \lambda_s$, then the following conditions are satisfied:
					\begin{enumerate}
					\renewcommand{\labelenumi}{(\roman{enumi})}
						\item $\operatorname{lcm}(\lambda_1,\lambda_2,\cdots, \hat{\lambda_i}, \cdots , \lambda_s) = M$ for all $i \in \{ 1,2,\ldots, s\}$, where $\hat{\lambda_i}$ denotes the omission of $\lambda_i$.
						\item $M$ divides $n$, and if $g' = 0$, then $M=n$.
						\item $s \neq 1$, and, if $g'=0$, then $s \geq 3$.
					\end{enumerate}
		\end{prop}
		
	We will use the following characterization of irreducibility of finite groups actions due to Gilman.
		
		\begin{thm}[{Gilman \cite[Lemma 3.9]{Gilman}}]
		\label{thm:Gilman}
			Let $G$ be a finite subgroup of $\operatorname{Diff}_+(\Sigma_g)$.
			Then $G$ is irreducible if and only if $(\Sigma_g \backslash M_G) / G$ is homeomorphic to $S^2 \backslash \{3 \text{-points}\}$.
		\end{thm}
	
	
\section{Preliminaries}
    
	In this section, we will classify irreducible periodic diffeomorphisms which commutes with an involution $\iota_{g}$ such that $\Sigma_{g}/\iota_{g}$ is a torus to prove Theorem \ref{thm:irr1}.
	Firstly, let us compute the total valency of $h_{n,p}$ defined in the introduction.

		\begin{prop}
		\label{prop:val}
			The total valency of $h_{n,p}$ is $[\, g, n\, ; \, \frac{1}{n} + \frac{p}{n} + \frac{n-p-1}{n} \,]$, where $g=\frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}$.
		\end{prop}
		
		\begin{proof}
    		The genus $g$ of the surface can be computed by counting the edges of the $2n$-gon $P$ (see Figure \ref{fig:Sig}) as 
        		\[
    	        	g=\frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}.
        		\]
			Clearly $h_{n,p}$ has $3$ multiple orbits: The barycenter of $P$, the vertices of $P$ and the middle point of the edges of $P$.
			Thus, the total valency of $h_{n, p}$ is of the form $[ \,  g, n \,;\, \frac{\theta_1}{\lambda_1}+ \frac{\theta_2}{\lambda_2}+ \frac{\theta_3}{\lambda_3} \, ]$ for some coprime integer $\lambda_i, \theta_i$ such that $\lambda_i$ divides $n$ for $i = 1, 2, 3$.
			
			It is clear that the valency of the fixed point at the barycenter of the $2n$-gon is $\frac{1}{n}$.
			Hence, we can set $\frac{\theta_1}{\lambda_1} = \frac{1}{n}$.
			
			Let us compute $\frac{\theta_2}{\lambda_2}$ the valency of the multiple orbit which is consisted by the body-centered points of the barycenter of Figure \ref{fig:Sig}.
			
				\begin{figure}[ht]
				    \centering
			        \captionsetup{width=0.6\linewidth}
			        \includegraphics[width=0.45\textwidth]{Figures/Sigval.pdf}
				    \caption{The cyclic order of\\ fundamental domains of $h_{n,p}$ around $y$}
				    \label{fig:Sigval}
		        \end{figure}
		
			Set $k = \opn{gcd} (n, p)$.
			Let $n'$ and $p'$ be the integer such that $n = k n'$ and $p = k p'$, respectively.
			The period of the multiple orbit is $k$, and hence, $\lambda_2 = n'$.
			Take a point $y$ in the orbit, and consider a loop which goes around the boundary of a small disk centered at $y$ as shown in Figure \ref{fig:Sigval}.
			The fundamental domains appear along this loop in the order
				\[
					0, \, p, \, 2p, \ldots,  (n'-2)p, \, (n'-1)p
				\]
			modulo $n$.
			Then, there uniquely exists $\nu \in \left\{ 1, 2, \ldots, n' \right\}$ such that $\nu p \equiv k \, \mod n$.
			Hence, we have $\nu p' \equiv 1\, \mod n'$.
			By the definition of a valency, $\theta_2 = p'$, and hence, $\frac{\theta_2}{\lambda_2} = \frac{p'}{n'} = \frac{p}{n}$.
			
			By Proposition \ref{prop:Nielsenint}, the valency $\frac{\theta_3}{\lambda_3}$ of another multiple orbit holds the condition
				\[
					\frac{1}{n} + \frac{p}{n} + \frac{\theta_3}{\lambda_3} \in \mathbb{Z}.
				\]
					Thus, we have $\frac{\theta_3}{\lambda_3} = \frac{n-p-1}{n}$.
		\end{proof}

    Applying the Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we can obtain the following well-known classification of periodic diffeomorphisms on tori.

        \begin{prop}
        \label{prop:brto}
            Let $f$ be a nontrivial orientation preserving periodic diffeomorphism on a torus $T$.
            Then $T/\langle f \rangle$ is homeomorphic to either $T^{2}$ or $S^{2}$.
            
            If $T/\langle f \rangle \cong T^2$, then $f$ is multiple orbit free.
            If $T/\langle f \rangle \cong S^2$, then the total valency of $f$ is one of the following:
                    \begin{align*}
                    h_{4,1}^2 = h_{6,3}^3 = & \left[\,1,2\,;\, \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}\,\right], && \\
                    h_{4,1} = & \left[\,1,4\,;\, \frac{1}{4} + \frac{1}{4} + \frac{1}{2}\,\right], &
                    h_{4,1}^{3}= & \left[\,1,4\,;\, \frac{3}{4} + \frac{3}{4} + \frac{1}{2}\,\right], \\
                    h_{6,3} = & \left[\,1,6\,;\, \frac{1}{6} + \frac{1}{3} + \frac{1}{2}\,\right], &
                    h_{6,3}^{2}= & \left[\,1,3\,;\, \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\,\right], \\
                    h_{6,3}^{4}= & \left[\,1,3\,;\, \frac{2}{3} + \frac{2}{3} + \frac{2}{3}\,\right], &
                    h_{6,3}^{5}= & \left[\,1,6\,;\, \frac{5}{6} + \frac{2}{3} + \frac{1}{2}\,\right].
                \end{align*}
        \end{prop}
        
    It is easy to see that we can identify $h_{3,1}$ with $h_{6,3}^2$ by deforming the $12$-gon to define $h_{6,3}$ to a hexagon (see Figures \ref{fig:1232}, see also Lemma \ref{lem:Inv}(1)).
    
        \begin{figure}[ht]
        \begin{tabular}{cc}
            \centering
            \begin{minipage}{0.45\textwidth}
                \captionsetup{width=0.85\linewidth}
                \includegraphics[width=1\textwidth]{Figures/1232.pdf}
                \caption{$h_{6,3}^2$}
                \label{fig:1232}
            \end{minipage}\hfill
            \begin{minipage}{0.45\textwidth}
                \captionsetup{width=0.85\linewidth}
                \includegraphics[width=0.9\textwidth]{Figures/111.pdf}
                \caption{$h_{3,1}$}
                \label{fig:h31}
            \end{minipage}
        \end{tabular}
        \end{figure}

    Since any multiple point free periodic diffeomorphism on a torus is reducible, we get the consequence.
    
        \begin{prop}
            Every irreducible periodic diffeomorphism on a torus is conjugate to one of the following: 
                \begin{enumerate}
                    \item $h_{4,1}$ or its inverse, 
                    \item $h_{6,3}$ or its inverse,
                    \item $h_{3,1}$ or its inverse.
                \end{enumerate}
        \end{prop}

	Here $h_{4,1}$, $h_{6,3}$ and $h_{3,1}$ are shown in Figures \ref{fig:112}, \ref{fig:123} and \ref{fig:h31}, respectively.

		\begin{figure}[ht]
			\centering
			\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/112.pdf}
				\caption{$h_{4,1}$}
				\label{fig:112}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/12312.pdf}
				\caption{$h_{6,3}$}
				\label{fig:123}
			\end{minipage}
			\end{tabular}
		\end{figure}
		
\section{Cyclic branched coverings of tori}
    
	Take an involution $\iota_g$ of $\Sigma_g$ whose quotient space is homeomorphic to a torus.
	Let $f$ be a periodic diffeomorphism of $\Sigma_g$ of order $n$ which commutes with $\iota_g$.
	
	Let $\bar{f}$ be a periodic diffeomorphism of $\Sigma_{g} / \langle \iota_g \rangle$ induced by $f$.
	Let $n=\operatorname{ord}(f)$ and $\bar{n}=\operatorname{ord}(\bar{f})$.
	We have the following simple observation.

		\begin{lem}
		\label{lem:1}
			\begin{enumerate}
				\item We have $n = 2\bar{n}$ if and only if $f^{\bar{n}} = \iota_g$.
				\item We have $n = \bar{n}$ if and only if $f^{\bar{n}} \neq \iota_g$.
			\end{enumerate}
		\end{lem}
		
		\begin{proof}
			We have $\bar{f}^{\bar{n}} = \operatorname{id}_{\Sigma_{g}/\langle \iota_g \rangle}$.
			Since the lifts of $\operatorname{id}_{\Sigma_{g} / \langle \iota_g \rangle}$ to $\Sigma_{g}$ are $\iota_g$ and $\operatorname{id}_{\Sigma_{g}}$, we have either $f^{\bar{n}} = \iota_g$ or $f^{\bar{n}} = \operatorname{id}_{\Sigma_{g}}$.
			The latter is equivalent to $n = \bar{n}$, and the former implies $n = 2\bar{n}$.
			The proof is concluded.
		\end{proof}
		
	We will use the following well known result.
		
		\begin{lem}
        \label{lem:fix3}
            There exists no cyclic branched covering $p : S^{2} \to S^{2}$ which has more than two branched points $x_{1}, x_{2}, \dots, x_{s} \in S^{2}$ such that $p^{-1}(x_{i})$ is a one point set for $i=1,2,3$.
            In particular, there exists no nontrivial periodic diffeomorphism of an orbifold homeomorphic to $S^{2}$ which fixes more than two points.
        \end{lem}
        
        \begin{proof}
            Assume that there exists such an $n$-fold branched covering $p : S^{2} \to S^{2}$. 
            By Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we have
                \[
                    -2 = n \left( -2 + 3 \left( 1 - \frac{1}{n} \right) + \sum_{i=4}^{s}\left( 1 - \frac{1}{\lambda_{i}} \right) \right),
                \]
            where $\lambda_{i}$ is the ramification index of $x_{i}$ for $i=4, \dots, s$. Then we have 
                \[
                     1-n = \sum_{i=4}^{s}\left( 1 - \frac{1}{\lambda_{i}} \right) \geq 0,
                \]
            which implies $n \leq 1$, and hence a contradiction.
        \end{proof}
	
	Let $G$ be the group generated by $f$ and $\iota_g$. 
	In this section, we will classify the $G$-action up to conjugacy to prove Theorem \ref{thm:irr1}.
	
	The following simple observation is fundamental. 
	
		\begin{lem}
		\label{lem:pres}
		    \begin{enumerate}
		        \item We have $f \left(\operatorname{Fix} (\iota_{g}) \right) = \operatorname{Fix} (\iota_{g})$, where $\operatorname{Fix} (\iota_{g})$ is the fixed point set of $\iota_g$.
		        \item The involution $\iota_g$ maps every $f$-orbit to an $f$-orbit of the same valency.
		    \end{enumerate}
		\end{lem}
		
    	\begin{proof}
        	It is easy to see that the valency of an $f$-orbit of $x$ is determined by the isotropy action of $f^{k}$, where $k$ is the period of $x$.
        	The lemma follows from the conjugation invariance of the isotropy action.
	    \end{proof}
	
	It is easy to determine the conjugacy class of the $G$-action in the case where $G$ is cyclic. Let $\bar{f}$ be the diffeomorphism on $\Sigma_g / \langle \iota_g \rangle$ induced by $f$.

	Let $p$ denote the double covering $\Sigma_g \to \Sigma_g / \langle \iota_g \rangle$. In the sequel, we will use the following criterion.
	
		\begin{lem}
		\label{lem:brodd}
			If $p$ is ramified at a multiple point $x$ of $f$ of period $k$ such that $\frac{n}{k}$ is even, then $G$ is cyclic.
		\end{lem}
		
		\begin{proof}
		    By assumption, the isotropy group $G_{x}$ of $x$ is generated by $f^{k}$ and $\iota_{g}$.
		    Since $\frac{n}{k}$ is even, the subgroup of $G_{x}$ generated by $f^{k}$ is of even order.
		    Thus, it contains $f^{\frac{n}{2}}$, which is a $\pi$-rotation around $x$.
		    Therefore, it follows that $f^{\frac{n}{2}} = \iota_{g}$, which implies that
		    $G$ is cyclic.
		\end{proof}
	
	It is easy to see that if $\bar{f}$ is reducible then $G$ is reducible: 
	Indeed, if $\bar{f}$ preserves a system of isotopy classes of essential simple closed curves, then $G$ preserves its inverse image of $p$. 
	Hence, to classify irreducible group $G$, it is sufficient to consider the case where $\bar{f}$ is irreducible.

		\begin{figure}[ht]
			\centering
				\includegraphics[width=0.45\textwidth]{Figures/simplebr.pdf}
				\caption{A simple closed curve $\mathcal{C}$}
				\label{fig:simplebr}
		\end{figure}
		
		\begin{lem}
		\label{lem:freered}
    		Assume that $\bar{f}$ is conjugate to a power of $h_{n,p}$.	
	    	If the canonical projection $p : \Sigma_g \to \Sigma_g / \langle \iota_g \rangle$ is branched at a free orbit of $\bar{f}$, then $G$ is reducible.
		\end{lem}
		
		\begin{proof}
			Assume that $p$ is branched at a simple point $y$ of $\bar{f}$.
			Take an $\bar{f}$-invariant simple closed curve $\mathcal{C}$ of $\Sigma_g / \langle \iota_g \rangle$ so that it bounds a disk which contains the $\bar{f}$-orbit of $y$ as shown in Figure \ref{fig:simplebr}.
			Then the preimage $p^{-1}(\mathcal{C})$ is essential and preserved by $f$ and $\iota_g$, hence $G$ is reducible.
		\end{proof}
	
	Now, we will determine irreducible $f$ that induces $h_{4,1}$ on the torus $\Sigma_{g} / \langle \iota_g \rangle$.

	Since $\iota_g$ has $2g-2$ fixed points, $p$ is branched at $2g-2$ points.
	By Lemma \ref{lem:pres}, $\bar{f}$ maps the branched locus of $p$ to itself.
	By classifying nonempty $\bar{f}$-invariant sets with an even number of points in the union of the multiple orbits of $\bar{f}$, we obtain the following lemma.
	
		\begin{lem}
		\label{lem:4}
			If $\bar{f}=h_{4,1}$, then the branched locus $B_{p}$ of $p$ is one of the following:
				\begin{enumerate}
					\item $\{ x, y \}$,\label{case:31}
					\item $\{ z_1, z_2 \}$,\label{case:32}
					\item $\{ x, y, z_1, z_2 \}$,\label{case:33}
				\end{enumerate}
			where $x, y, z_1, z_2 \in \Sigma_{g}/\langle \iota_g \rangle$ are as shown in Figure \ref{fig:112}.
		\end{lem}
		
		\begin{prop}
		\label{prop:1}
			If $\bar{f}=h_{4,1}$, then $G$ is conjugate to either of $\langle h_{8,1} \rangle$ or $\langle h_{8,5} \rangle$.
		\end{prop}
		
		\begin{proof}
		    We can see that $G$ is cyclic in all three cases in Lemma \ref{lem:4}.
		    Indeed, since $\operatorname{ord} (\bar{f}) = 4$, and the period of branch points is either $1$ or $2$, we can see that the cardinality of the isotropy group of every branch point is even.
		    Then, by Lemma \ref{lem:brodd}, $G$ is cyclic.
		    By Lemma \ref{lem:1}, we have that $n=8$. 
		
			Let us consider the case \ref{case:31} in Lemma \ref{lem:4}.
			We can see that $p$ has three branched points of index $8,8$ and $2$, respectively.
			Then the total valency of $f$ is of the form $[\,2,8\, ; \, \frac{\theta_1}{8}+ \frac{\theta_2}{8}+ \frac{1}{2} \,]$, where $\theta_1, \theta_2 \in \{ 1,3,5,7 \}$.
			Here $\frac{\theta_1}{8} + \frac{\theta_2}{8} + \frac{1}{2}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}), and hence, $(\theta_1, \theta_2) = (1,3)$ or $(5,7)$.
			In both cases, since $f^4$ is a hyperelliptic involution, it contradicts Lemma \ref{lem:1}.
			Thus, this case does not occur.
		
			In the case \ref{case:32} of Lemma \ref{lem:4}, we can see that $p$ has three branched points of index $4$.
			Thus the total valency of $f$ is of the form $[\,2,8\, ; \, \frac{\theta_1}{4}+ \frac{\theta_2}{4}+ \frac{\theta_3}{4} \,]$, where $\theta_{1}, \theta_{2}, \theta_{3} \in \{1,3\}$.
			It is clear that there are no $\theta_{1}, \theta_{2}, \theta_{3}$ such that $\frac{\theta_1}{4}+ \frac{\theta_2}{4}+ \frac{\theta_3}{4}$ is an integer.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), this case does not occur.
		
			Finally, let us consider the case \ref{case:33} of Lemma \ref{lem:4}.
			We can see that $p$ has three branched points of index $8,8$ and $4$, respectively.
			Let $[\,2,8\, ; \, \frac{\theta_1}{8}+ \frac{\theta_2}{8}+ \frac{\theta_3}{4} \,]$ be the total valency of $f$, where $\theta_1, \theta_2 \in \{ 1,3,5,7 \}$ and $\theta_3 \in \{1,3\}$.
			Now $\frac{\theta_1}{8} + \frac{\theta_2}{8} + \frac{\theta_3}{4}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}), and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,3k)$ or $(k,5k,k) \mod 8$ for $k \in \{ 1,3,5,7 \}$.
			Thus, $f$ is a power of either of $h_{8,1}=[\, 3,8 \,;\, \frac{1}{8}+\frac{1}{8}+\frac{3}{4} \,]$ or $h_{8,5}=[\, 3,8 \,;\, \frac{1}{8}+\frac{5}{8} +\frac{1}{4} \,]$.
		\end{proof}

    Let us consider the case where $\bar{f} = h_{6,3}$.
    We obtain the following lemma by an argument similar to the case where $\bar{f} = h_{4,1}$.

		\begin{lem}
		\label{lem:5}
			If $\bar{f}=h_{6,3}$, then branched locus $B_p$ of $p$ is one of the following:
				\begin{enumerate}
					\item $\{ y_1, y_2 \}$,\label{case:41}
					\item $\{ x, z_1, z_2, z_3 \}$,\label{case:42}
					\item $\{ x, y_1, y_2, z_1, z_2, z_3 \}$,\label{case:43}
				\end{enumerate}
			where $x, y_1, y_2, z_1, z_2, z_3 \in \Sigma/ \langle \iota_g \rangle$ are as shown in Figure \ref{fig:123}.
		\end{lem}
		
		\begin{prop}
		\label{prop:2}
			If $\bar{f}=h_{6,3}$, then $G$ is conjugate to a subgroup of either of $\langle h_{6,1}, I \rangle, \langle h_{12,2} \rangle$, or $\langle h_{12,3} \rangle$.
		\end{prop}
		
		\begin{proof}
		 	Let us consider the case \ref{case:42} of Lemma \ref{lem:5}.
		 	Since $p$ is branched at a fixed point $x$ of $\bar{f}$ and the order of $f$ is even, by Lemma \ref{lem:brodd}, $G$ is cyclic.
			Then, by Lemma \ref{lem:1}, we have $n=12$ and the branch indices of $f$ are $12,4$ and $3$.
			Let $[\,3,12\, ; \, \frac{\theta_1}{12}+ \frac{\theta_2}{4}+ \frac{\theta_{3}}{3} \,]$ be the total valency of $f$, where $\theta_1 \in \{ 1,5,7,11 \}, \theta_2 \in \{ 1,3 \}$ and $\theta_3 \in \{1,2 \}$.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta_1}{12} + \frac{\theta_2}{4} + \frac{\theta_3}{3}$ is an integer, and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,2k) \mod 12$ for $k \in \{ 1,5,7,11 \}$.
			Thus, $f$ is a power of $h_{12,3}=[\, 3,12 \,;\, \frac{1}{12}+\frac{1}{4}+\frac{2}{3} \,]$ shown in Figure \ref{fig:138}.
			Note that $h_{12,3}^6$ is equal to $\iota_3$.
			
			Let us consider the case \ref{case:43} of Lemma \ref{lem:5}.
			Since $p$ is branched at a fixed point $x$ of $\bar{f}$ and the order of $f$ is even, by Lemma \ref{lem:brodd}, $G$ is cyclic.
			Then, by Lemma \ref{lem:1}, we have $n=12$, and the branch indices of $f$ are $12,6$ and $4$.
			Let $[\,3,12\, ; \, \frac{\theta_1}{12}+ \frac{\theta_2}{6}+ \frac{\theta_{3}}{4} \,]$ be the total valency of $f$, where $\theta_1 \in \{ 1,5,7,11 \}, \theta_2 \in \{ 1,5 \}$ and $\theta_3 \in \{1,3 \}$.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta_1}{12} + \frac{\theta_2}{6} + \frac{\theta_3}{4}$ is an integer, and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,3k) \mod 12$ for $k \in \{ 1,5,7,11 \}$.
			Thus, $f$ is a power of $h_{12,2}=[\, 3,12 \,;\, \frac{1}{12}+\frac{1}{6}+\frac{3}{4} \,]$.
			Note that $h_{12,2}^{6}$ is equal to $\iota_4$.
			
			Finally, let us consider the case \ref{case:41} in Lemma \ref{lem:5}.
	    	Since $p$ is branched at $2$ points, we have $g=2$.
			Since $\bar{f}$ fixes $x$, it follows that $p^{-1}(x)$ is either a multiple orbit of period two, or $f$ fixes $p^{-1}(x)$ pointwisely.
			In the former case, $\iota_{g} \circ f$ is a lift of $\bar{f}$ which fixes $p^{-1}(x)$ pointwisely.
			Thus, to classify the conjugacy class of $G$, we can assume that $f$ fixes $p^{-1}(x)$ pointwisely without loss of generality.
			In this case, we can see that $n=6$.
			Since the valency of $\{y_{1},y_{2}\}$ as an $\bar{f}$-orbit is $\frac{1}{3}$, by a local argument, we can see that the valency of $p^{-1}(\{y_1, y_2 \})$ is either $\frac{2}{3}$ or $\frac{1}{6}$.
			But $n=6$ yields that only the former case occurs.
			Similarly, $p^{-1}(\{ z_1, z_2, z_3 \})$ is either a free orbit or the union of two multiple orbits of valency $\frac{1}{2}$.
		 	The Riemann-Hurwitz formula (Proposition \ref{prop:RH}) yields that only the former case occurs here.
		 	Indeed, if we substitute $g=2$ and $(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}) = (6,6,3,2,2)$ to the formula,
			then we get $g' = -\frac{1}{2}$, which contradicts to the fact that $g'$ is the genus of $\Sigma_{2}/\langle f \rangle$.
			Thus, it follows that the total valency of $f$ is $[\,2,6\,;\, \frac{1}{6}+\frac{1}{6}+\frac{2}{3} \,]$.
			Thus $f$ is conjugate to $h_{6,1}$.
			Now $f^{3} \circ \iota_{2}$ is a hyperelliptic involution which fixes all points in $p^{-1}(\{ z_1, z_2, z_3 \})$.
			By \cite[Lemma 13]{TN}, $G$ is conjugate to $\langle h_{6,1}, I \rangle$, where $I$ is a hyperelliptic involution on $\Sigma_{2}$.
		\end{proof}

    Finally, let us consider the case where $\bar{f} = h_{3,1}$.
    The multiple orbits of $h_{3,1}$ are three fixed points.
    A nonempty $\bar{f}$-invariant subset with an even number of points in the set of these three fixed points is a two point set.
    All of them are mutually conjugate, and hence we can assume that $p$ is branched at $y$ and $z$ in Figure \ref{fig:h31}.
    Thus, the branch locus of $p$ is the same as the case the case \ref{case:41} in Lemma \ref{lem:5}.
    By a parallel argument, we can see that $G$ is conjugate to a subgroup of $\langle h_{6,1}, I \rangle$.

    Theorem \ref{thm:irr1} follows from Propositions \ref{prop:1}, \ref{prop:2} and the argument in the last paragraph.

\section{Finite subgroups of the centralizer of irreducible periodic diffeomorphisms on surfaces}


    In this chapter, we will classify up to conjugacy finite subgroups of the centralizer of irreducible periodic diffeomorphisms to prove Theorem \ref{thm:irr2}.

    Let $f$ be an irreducible periodic diffeomorphism of $\Sigma_g$ of period $n$.
	By Theorem \ref{thm:Gilman}, the total valency of $f$ is of the form $[\,g,n\,;\, \frac{\theta_1}{n} + \frac{\theta_2}{n} + \frac{\theta_3}{n} \,]$ for some $\theta_{1}$, $\theta_{2}$, $\theta_{3}$ such that $1 \leq \theta_{i} \leq \frac{n}{2}$ for $i=1,2,3$.
	Let $F$ be the subgroup of $\operatorname{Diff}_+ (\Sigma_g)$ generated by $f$, and let $H$ be a finite subgroup in the centralizer of $f$ in $\operatorname{Diff}_+ (\Sigma_g)$.

    	\begin{lem}
		\label{lem:diff}
			If $\theta_1, \theta_2$ and $\theta_3$ are mutually different, then $H$ is a subgroup of $F$.
		\end{lem}
		
		\begin{proof}
	    	Take an arbitrary $d \in H$.
	    	The diffeomorphism $d$ maps a multiple orbit of $f$ to another multiple orbit of the same valency by Lemma \ref{lem:pres}.
	    	Therefore, by assumption, $d$ maps each multiple orbit of $f$ to itself.
	    	Then $d$ induces $\bar{d}$ on $\Sigma/ F \cong S^2$, which fixes $3$ cone points.
	    	By Lemma \ref{lem:fix3}, it follows that $\bar{d}=\operatorname{id}_{\Sigma_g / F}$.
			Since any lift of $\operatorname{id}_{\Sigma_g / F}$ is of the form $f^k$ for some integer $k$, we have $d=f^{k}$.
			Therefore $H$ is a subgroup of $F$.
		\end{proof}
	
	Now let us consider the case where $\theta_{1} = \theta_{2}$.
	We observe the following.
	
	    \begin{lem}\label{lem:g1}
	        If $\theta_{1} = \theta_{2} = \theta_{3}$, then $f$ is conjugate to $h_{3,1}$ or $h_{3,1}^{2}$.
	        In particular, we have $g=1$.
        \end{lem}

        \begin{proof}
            Let $\theta=\theta_{1}=\theta_{2}=\theta_{3}$.
            Here $\frac{3\theta}{n}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}).
            Since $0 < \frac{\theta}{n} < 1$, we have $\frac{3\theta}{n}=1$ or $2$.
            In the case where $\frac{3\theta}{n}=1$, we have $(n,\theta) = (3,1)$.
            Otherwise we have $\frac{3\theta}{n}=2$, and hence $(n,\theta) = (3,2)$.
            In both cases, by the Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we have $g=1$. Thus the total valency of $f$ is equal to either $[\, 1,3\,;\frac{1}{3}+\frac{1}{3}+\frac{1}{3} \,]$ or $[\,1,3\,;\frac{2}{3}+\frac{2}{3}+\frac{2}{3}\,]$.
            The former is the total valency of $h_{3,1}$, and the latter is the total valency of $h_{3,1}^{2}$.
            Thus, $f$ is conjugate to $h_{3,1}$ or $h_{3,1}^{2}$ by Theorem \ref{thm:Nielsen}.
        \end{proof}

    Thus, in order to prove Theorem \ref{thm:irr2}, it is sufficient to consider the case where
        \[
            \theta_1=\theta_2\neq \theta_3.
        \]
    By the first condition of Proposition \ref{prop:Harvey}, $\operatorname{gcd}(\theta_1,n)=1$, and hence, $f$ is conjugate to the $\theta_1$-th power of $h_{n,1}$.
    Clearly $H$ is conjugate to a finite subgroup of the centralizer of $h_{n,1}$. Thus, we can assume that $f=h_{n,1}$ and $H$ is a finite subgroup of the centralizer of $h_{n,1}$ in the following. Since $g = \frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, n-p-1) + 1}{2}$, we can compute the period $n$ in terms of the genus $g$ as follows:
        \[
            n = \begin{cases}
                    2g+1 & n : \text{odd,} \\
                    2g+2 & n : \text{even.}
                 \end{cases}
        \]
    
    Since $[h_{n,1}]=[\,g,n\,;\,\frac{1}{n}+\frac{1}{n}+\frac{n-2}{n}\,]$, the diffeomorphism $h_{n,1}$ have two fixed points of valency $\frac{1}{n}$ and a multiple orbit of valency $\frac{n-2}{n}$, which is a fixed point if $n$ is odd and has period $2$ if $n$ is even.
    We have, for every $n$, a (hyperelliptic) involution which commutes with $h_{n,1}$ and exchanges two fixed points of valency $\frac{1}{n}$ as follows:
    
        \begin{lem}
        \label{lem:Inv}
            \begin{enumerate}
                \item[ ] 
                \item $h_{4g+2,2g+1}^{2g}$ is conjugate to $h_{2g+1,1}$, and $h_{4g+2,2g+1}^{2g+1}$ is a hyperelliptic involution which commutes with $h_{4g+2,2g+1}^{2g}$ and exchanges two fixed points of $h_{4g+2,2g+1}^{2g}$ of valency $\frac{1}{2g+1}$.
                \item $I$ shown in Figure \ref{fig:Iphi3} is a hyperelliptic involution which commutes with $h_{2g+2,1}$ and exchanges two fixed points of $h_{2g+2,1}$ of valency $\frac{1}{2g+2}$.
           \end{enumerate}
        \end{lem}

        \begin{proof}
            By Proposition \ref{prop:val}, the total valency of $h_{4g+2,2g+1}$ is equal to $[\,g,4g+2\,;\, \frac{1}{4g+2} + \frac{g}{2g+1} + \frac{1}{2} \,]$.
            Since $\operatorname{gcd} (4g+2, 2g)=2$, the period of $h_{4g+2,2g+1}^{2g}$ is $2g+1$.
            Figure \ref{fig:h4g+2} is a decomposition of $\Sigma_g$ into the fundamental domain of $h_{4g+2,2g+1}$.
            
            The multiple orbit of $h_{4g+2,2g+1}$ with valency $\frac{1}{2}$ consists of $2g+1$ points, and it is a free orbit of $h_{4g+2,2g+1}^{2g}$.
            
            By construction, $h_{4g+2,2g+1}$ is induced from the clockwise $\frac{2\pi}{4g+2}$-rotation the $8g+4$-gon.
            Thus, $h_{4g+2,2g+1}^{2g}$ is induced from $\frac{2g\pi}{2g+1}$-rotation (See Figure \ref{fig:h2g+1}).
            Let $x$ be the barycenter of Figure \ref{fig:h4g+2}.
            Since we have
                \[
                    g (2g-1) \equiv -2g \equiv 1 \ \mod 2g+1,
                \]
            by the definition of valency, the valency of $x$ is equal to $\frac{2g-1}{2g+1}$.
            
                \begin{figure}[ht]
	        	\centering
	        	\begin{tabular}{cc}
	        	\begin{minipage}{0.45\textwidth}
			    	\centering
	        		\captionsetup{width=0.87\linewidth}
		            \includegraphics[width=0.94\textwidth]{Figures/h4g+2.pdf}
    				\caption{Fundamental domains of $h_{4g+2}$}
	    			\label{fig:h4g+2}
		    	\end{minipage}\hfill
			    \begin{minipage}{0.45\textwidth}
				    \centering
		    		\captionsetup{width=0.87\linewidth}
			    	\includegraphics[width=0.94\textwidth]{Figures/h2g+1.pdf}
				    \caption{Fundamental domains of $h_{4g+2,2g+1}^{2g}$}
			    	\label{fig:h2g+1}
			    \end{minipage}
		    	\end{tabular}
            	\end{figure}
            
            The period of the other multiple orbit of $h_{4g+2,2g+1}$ is $2$.
            Thus $h_{4g+2,2g+1}^{2g}$ fixes this orbit pointwisely.
            Let $\frac{\theta}{2g+1}$ be its valency.
            Then, by Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta}{2g+1}+\frac{\theta}{2g+1}+\frac{2g-1}{2g+1}$ is an integer, and hence, we have $\theta=1$.
            Therefore, we obtain 
                \[
                    [h_{4g+2,2g+1}^{2g}] = \left[\,g,2g+1\,\,;\,\frac{1}{2g+1}+\frac{1}{2g+1}+\frac{2g-1}{2g+1}\,\right] = [h_{2g+1,1}],
                \]
            which concludes the proof by Theorem \ref{thm:Nielsen}.
        \end{proof}
        
        \begin{lem}
            \begin{enumerate}
                \item $H$ is conjugate to a subgroup of $\langle h_{4g+2,2g+1} \rangle$, where $h_{2g+1,1}$ is identified with $h_{4g+2,2g+1}^{2g}$.
                \item $H$ is conjugate to a subgroup of $\langle h_{2g+2,1}, I\rangle$.
            \end{enumerate}
        \end{lem}

        \begin{proof}
            By Lemma \ref{lem:Inv}, for every $n$, we have an involution $I_{n}$ which commutes with $h_{n,1}$ and two fixed points of valency $\frac{1}{n}$.
            By Lemma \ref{lem:pres}, one of the following two cases occurs:
			
			    \begin{enumerate}
			        \item[(A)] $d$ maps each of three multiple orbits of $h_{n,1}$ to itself.
			        \item[(B)] $d$ exchanges two fixed points of $h_{n,1}$ with valency $\frac{1}{n}$, and maps the multiple orbit of $f$ with valency $\frac{n-2}{n}$ to itself.
			    \end{enumerate}

            It is easy to see that, by a similar argument to Lemma \ref{lem:diff}, we have $d \in F$ in the case (A).
            Thus, it is sufficient to consider the case where there exists $d \in H$ which satisfies the condition (B).
            Note that, for any $d' \in H$ which satisfies the condition (B),
            $d^{-1} \circ d'$ maps every three multiple orbits to itself.
            Then, Lemma \ref{lem:fix3} yields that $d^{-1} \circ d' = h_{n,1}^{k}$ for some $k$.
            Therefore $d' \in \langle h_{n,1}, d\rangle$.
            
            Consider the map $\bar{d}$ induced on $\Sigma_{g}/F$ by $d$.
            The map $I_{n}$ also induces a diffeomorphism $\bar{I}_{n}$ on $\Sigma_{g}/F$.
            Since both $\bar{d}$ and $\bar{I}_{n}$ are involutions which fix the same cone point,
            we have an orbifold diffeomorphism $\kappa : (\Sigma_{g}/F)/\bar{d} \to (\Sigma_{g}/F)/\bar{I}$.
            Then, by lifting $\kappa$ to a map $\Sigma_{g}/F \to \Sigma_{g}/F$, we have an orbifold diffeomorphism $\kappa_{1}$ which makes the following diagram commutative:
                \[
                    \xymatrix{
                    \Sigma_{g}/F \ar[r]^{\kappa_{1}} \ar[d] & \Sigma_{g}/F \ar[d] \\
                    (\Sigma_{g}/F)/\bar{d} \ar[r]_{\kappa} & (\Sigma_{g}/F)/\bar{I},
                    }
                \]
            where the vertical maps are the canonical projections.
            Then, since $\kappa$ is an isomorphism of double branched coverings, we have $\bar{d} \circ \kappa = \kappa \circ \bar{I}_{n}$.
            Therefore $\bar{d}$ is conjugate to $\bar{I}_{n}$.
            Then we may assume that $\bar{d} = \bar{I}_{n}$.
            Since $\bar{d} \circ \bar{I}_{n}^{-1} = \operatorname{id}_{\Sigma_{g}/F}$, we can see that $d \circ I_{n} \in F$.
            It follows that $d \in \langle h_{n,1}, I_{n} \rangle$.
        \end{proof}

    Theorem \ref{thm:irr2} follows from the last lemma and the preceding argument.


    We can see an exceptional phenomenon in the case of $g=1$.
    In this case, $f$ may be conjugate to $h_{3,1}$ or $h_{3,1}^{2}$ by Lemma \ref{lem:g1}.
    Since the centralizer of $f$ coincides up to conjugacy in these two cases, we can assume that $f=h_{3,1}$. As explained in the paragraph after Proposition \ref{prop:brto} (see Figures \ref{fig:1232}, \ref{fig:h31}), we can identify $h_{3,1}$ with $h_{6,3}^{2}$. Therefore $h_{6,3}$ commutes with $h_{3,1}$.
    In addition to $h_{6,3}$, we have one more periodic diffeomorphism $J$ which commutes with $h_{3,1}$ as shown in Figure \ref{fig:111J}.
    This $J$ and $h_{6,3}$ generates a finite group in the centralizer of $h_{3,1}$.

    \begin{figure}[ht]
	        \begin{tabular}{cc}
	            \begin{minipage}{0.44\textwidth}
	                \centering
	    		    \captionsetup{width=0.85\linewidth}
		        	\includegraphics[width=0.85\textwidth]{Figures/1111.pdf}
	            \end{minipage}\hfill \hspace{5pt}$\longrightarrow$
	       \hspace{2pt}
	       \begin{minipage}{0.44\textwidth}
	                \centering
	    		    \captionsetup{width=0.8\linewidth}
		        	\includegraphics[width=0.85\textwidth]{Figures/1112.pdf}
	            \end{minipage}
	        \end{tabular}
	        \caption{A periodic diffeomorphism $J$ which commutes with $h_{3,1}$}
	        \label{fig:111J}
	    \end{figure}
    
\input{references}
\end{document}



\section{automorphism of the Riemann sphere}

    In this section, we will obtain the Lemma \ref{} which is often used in this paper.
    
    Since any sphere is conformally equivalent to the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C}\ ^{\cup} \{ \infty \}$, we will consider the automorphism of $\hat{\mathbb{C}}$.
    
    Firstly, we have the following fact.
    
        \begin{prop}
            For arbitrary automorphism $f$ of $\hat{\mathbb{C}}$, there exist complex numbers $a,b,c,d$ such that $ad-bc \neq 0$ and $f(Z) = \frac{az+b}{cz+d}$.
        \end{prop}
        
        \begin{prop}
            Take arbitrary mutually distinct points $z_1,z_2,z_3$ of $\hat{\mathbb{C}}$.
            For mutually distinct points $w_1,w_2,w_3$, there uniquely exists the automorphism $f$ such that $f(z_i) = w_i$.
        \end{prop}
        
        \begin{lem}
            If an automorphism of $S^2$ fixes mutually distinct three points, then the automorphism is $\operatorname{id}_{S^2}$.
        \end{lem}
        
        \begin{proof}
            Take an automorphism $f$ which fixes mutually distinct points $z_1, z_2, z_3$.
            By Proposition \ref{}, there uniquely exists the automorphism $g$ such that, $g(z_1)=0, g(z_2)=1$ and $g(z_3)= \infty$.
            Let us consider the automorphism $h = g\circ f \circ g^{-1}$.
            By Proposition \ref{}, there exists complex numbers $a,b,c,d$ such that $ad-bc \neq 0$ and $h(z) = \frac{az+b}{cz+d}$.
            Since $h$ fixes $0,1$ and $\infty$, we have $\frac{b}{d} = 0, \frac{a+b}{c+d} = 1$ and $\frac{a}{c} = \infty$.
            This implies that $b=c=0$ and $a=d$, and hence, $h= \operatorname{id}_{\hat{\mathbb{C}}}$.
            Thus, we have $f = g^{-1} h g = \operatorname{id}_{\hat{\mathbb{C}}}$.
            This completes the proof of Lemma \ref{}.
        \end{proof}



\section{Periodic diffeomorphisms of tori}

	We classify actions of finite abelian groups on $T^{2}$ by using the well-known classification of wall paper groups.
	Let $G$ be a finite abelian subgroup of $\operatorname{Diff}_+(T^2)$.
	Since $\mathbb{C}$ is the universal covering space of $T^2$ and $G$ preserves the orientation of $T^2$, we can lift $G$ to one of the wallpaper groups $p1, p2, p3, p4$ and $p6$ as shown in Figures \ref{fig:p1}, \ref{fig:p2}, \ref{fig:p3}, \ref{fig:p4} and \ref{fig:p6}, respectively.
		
		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p1.pdf}
				\caption{}
				\label{fig:p1}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p2.pdf}
				\caption{}
				\label{fig:p2}
			\end{minipage}\\
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p3.pdf}
				\caption{}
				\label{fig:p3}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p4.pdf}
				\caption{}
				\label{fig:p4}
			\end{minipage}\\
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=0.9\textwidth]{p6.pdf}
				\caption{}
				\label{fig:p6}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
			\end{minipage}
		\end{tabular}
		\end{figure}
	
	By projecting their group action to $T^2$, the following proposition holds.
	
		\begin{prop}
		\label{prop:torus}
			$G$ is conjugate to a subgroup of a group generated by one of the following generators.
				\begin{enumerate}
					\item $2$ translations as shown in Figure \ref{fig:pp1},
					\item $h_{4,1}^2$ as shown in Figure \ref{fig:pp2},
					\item $h_{3,1}$ and a translation as shown in Figure \ref{fig:111J},
					\item $h_{4,1}$ and a hyperelliptic involution $I$ as shown in Figure \ref{fig:112I},
					\item $h_{6,3}$ as shown in Figure \ref{fig:pp6}.
				\end{enumerate}
		\end{prop}
		
		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{pp2.pdf}
				\caption{}
				\label{fig:pp2}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{123.pdf}
				\caption{}
				\label{fig:pp6}
			\end{minipage}
		\end{tabular}
		\end{figure}
		
	Since $\iota_1$ is a translation on $T^2$, the following holds. 
		
		\begin{cor}
		\label{cor:torus}
		    Any finite abelian subgroup of $\operatorname{Diff}_+(\Sigma_g)$ which contains $\iota_1$ is conjugate to a subgroup of Figures \ref{fig:pp1}, \ref{fig:112I} or \ref{fig:111J}.
		\end{cor}
\section{Introduction}
    
    An involution on an oriented closed surface $\Sigma_{g}$ of genus $g>1$ is called \emph{hyperelliptic} if it fixes $2g+2$ points.
    It is well known that such involution is unique up to conjugacy, and has the maximum number of fixed points among involutions on $\Sigma_{g}$.
    A diffeomorphism of $\Sigma_{g}$ which commutes with a hyperelliptic involution is called \emph{hyperelliptic}.
    Ishizaka \cite{Ishizaka3,Ishizaka} classified hyperelliptic periodic diffeomorphisms up to conjugacy and gave the right-handed Dehn twist presentations of their mapping classes based on resolutions of singularities of families of Riemann surfaces.
    An involution $\iota$ of $\Sigma_{g}$ is hyperelliptic if and only if the quotient space $\Sigma_{g}/\langle\iota\rangle$ is homeomorphic to a sphere.
    In this paper, we will classify periodic diffeomorphisms of $\Sigma_{g}$ which commute with an involution $\iota_{g}$ whose quotient space $\Sigma_{g}/\langle \iota_{g} \rangle$ is homeomorphic to a torus.
    In this case, there is a significant difference in the classification from the hyperelliptic case considered in \cite{Ishizaka}.
    Recall that a diffeomorphism of $\Sigma_g$ is called {\it reducible} if there exists a family of isotopy classes of essential simple closed curves on $\Sigma_g$ preserved by the diffeomorphism and {\it irreducible} if otherwise.
    By the classification in \cite{Ishizaka}, every hyperelliptic periodic diffeomorphism of $\Sigma_{g}$ is a power of an irreducible periodic diffeomorphism.
    In turn, as we will see below, this is not the case for periodic diffeomorphisms which commute with an involution $\iota_{g}$ such that $\Sigma_{g}/\langle \iota_{g} \rangle \cong T^2$.
    (see Example \ref{ex:red}).
    In this article, we focus on the classification in the irreducible case.
    Irreducible periodic diffeomorphisms of $\Sigma_g$ have been studied from various viewpoints (see e.g. \cite{Hirose2,Hirose-Kasahara,Ge-Ra,Dhanwani}).

	To state the result, let us present an example of an irreducible periodic diffeomorphism $h_{n, p}$ of an oriented closed surface parametrized by a positive integer $n \geq 3$ and $p \in \{ 1,\ldots, n-1 \}$.
	Consider a regular $2n$-gon.
	Let $\alpha_i$ and $\beta_i\  ( i = 0, \ldots, n-1 )$ be the edges of the $2n$-gon shown in Figure \ref{fig:Sig}.
	By identifying $\alpha_i$ with $\beta_j$ for every pair of $i$ and $j \in \{ 0, \ldots, n-1 \}$ with $i-j \equiv p \mod n$, we obtain a surface $\Sigma$ of genus $g = \frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}$.
	We denote $h_{n,p}$ the diffeomorphism of $\Sigma$ induced from the clockwise $\frac{2\pi}{n}$-rotation of the $2n$-gon. To simplify the figures, both $\alpha_{0}$ and $\beta_{p}$ will be denoted by $\alpha$, and omit the notation of other edges in the rest of the paper.

		\begin{figure}[ht]
		\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				 
				  	\includegraphics[width=0.9\textwidth]{Figures/Sig.pdf}
				\caption{A diffeomorphism $h_{n,p}$}
				\label{fig:Sig}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\captionsetup{width=.8\linewidth}
				\includegraphics[width=0.9\textwidth]{Figures/connsum.pdf}
				\caption{Connected sum of some irreducible periodic diffeomorphisms}
				\label{fig:connsum}
			\end{minipage}
		\end{tabular}
		\end{figure}

        \begin{ex}
        \label{ex:red}
            We can construct an example of a reducible periodic diffeomorphism which commutes with an involution $\iota_{2g+1}$ from $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$ by a connected sum (Figure \ref{fig:connsum}):
            Remove four small disks centered at the fixed points of $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$, respectively, and glue the complements of the disks along the circle boundaries so that $h_{4g, 2g}$ and $h_{4g, 2g}^{4g-1}$ coincide with each other on the boundary.
            Then we get a periodic diffeomorphism $f$ on $(\Sigma_g - disks) \cup_{\partial} (\Sigma_g - disks)$ of period $4g$.
            Note that $f^{2g}$ is an involution whose quotient space is a torus which commutes with $f$.
        \end{ex}
    
    Let us state the main result of this article.
    In this result, we classify irreducible periodic diffeomorphisms which commutes with $\iota_g$. 
	
		\begin{thm}
		\label{thm:irr1}
			Let $\iota_g$ be an involution of $\Sigma_g$ such that $\Sigma_g/ \langle \iota_g \rangle$ is homeomorphic to a torus as shown in Figure \ref{fig:iotag}.
			For any periodic diffeomorphism $f$ of $\Sigma_{g}$ which commutes with an involution $\iota_g$, if the subgroup $G=\langle f, \iota_g \rangle$ of $\operatorname{Diff}_{+}(\Sigma_{g})$ is irreducible, then $G$ is conjugate to a subgroup of one of following:
			    \begin{enumerate}
			        \item $\langle h_{6,1}, I \rangle$ (Figure \ref{fig:114}),
			        \item $\langle h_{8,1} \rangle$ (Figure \ref{fig:116}),
			        \item $\langle h_{8,5} \rangle$ (Figure \ref{fig:125}),
			        \item $\langle h_{12,2} \rangle$ (Figure \ref{fig:129}),
			        \item $\langle h_{12,3} \rangle$ (Figure \ref{fig:138}).
			    \end{enumerate}
		\end{thm}
		
			It is remarkable that there are only finite cases in the classification of Theorem \ref{thm:irr1}.
			It is in contrast to the case of the classification of hyperelliptic periodic classes, where we have three infinite families of conjugacy classes. 
	
	
        \begin{figure}[ht]
	    	\centering
		    \includegraphics[scale=0.45]{Figures/iotag.pdf}
    		\caption{The involution $\iota_{g}$}
	    	\label{fig:iotag}
        \end{figure}
        
		\begin{figure}[ht]
		\begin{tabular}{ccc}
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/114.pdf}
				\caption{$h_{6,1}$}
				\label{fig:114}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/116.pdf}
				\caption{$h_{8,1}$}
				\label{fig:116}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/125.pdf}
				\caption{$h_{8,5}$}
				\label{fig:125}
			\end{minipage}\\
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/129.pdf}
				\caption{$h_{12,2}$}
				\label{fig:129}
			\end{minipage} \hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/138.pdf}
				\caption{$h_{12,3}$}
				\label{fig:138}
			\end{minipage}\hfill
			\begin{minipage}{0.33\textwidth}
				\centering
				\captionsetup{width=\linewidth}
				\includegraphics[width=1\textwidth]{Figures/Iphi3.pdf}
				\caption{$h_{2g+2,1}$ and $I$}
				\label{fig:Iphi3}
			\end{minipage}
		\end{tabular}
		\end{figure}

	
	For hyperelliptic involution $I$ and hyperelliptic periodic diffeomorphism $f$, the group $G = \langle f, I \rangle$ is cyclic except the case where $f$ is a power of $h_{2g+2,1}$ (see \cite{TN, Ishizaka}).
	Motivated by this fact, we give a classification of the centralizer of irreducible diffeomorphisms as follows.

        \begin{thm}
		\label{thm:irr2}
    		Let $f$ be an irreducible periodic diffeomorphism of $\Sigma_{g}$ with $g>1$, and $H$ a finite subgroup of the centralizer of $f$ in $\operatorname{Diff}_{+}(\Sigma_{g})$.
    		Then one of the following occurs:
    		
                \begin{enumerate}
                    \item $H$ is a subgroup of $\langle f \rangle$.
                    \item $H$ is conjugate to a subgroup of $\langle h_{4g+2,2g+1} \rangle$ so that $\langle f \rangle$ is conjugate to $\langle h_{2g+1,1} \rangle$.\label{case:irr22}
                    \item $H$ is conjugate to a subgroup of $\langle h_{2g+2, 1}, I \rangle$ so that $\langle f \rangle$ is conjugate to $\langle h_{2g+2,1} \rangle$.
                \end{enumerate}
	   	\end{thm}
		
	As a corollary, we obtain the following characterization of $h_{n,1}$.
		
		\begin{cor}
    		If an irreducible periodic diffeomorphism $f$ on $\Sigma_{g}$ commutes with two involutions $I_1$ and $I_2$ which are not mutually conjugate, then $f$ is conjugate to a power of either $h_{2g+1,1}$ or $h_{2g+2,1}$.   
		\end{cor}


\section{periodic diffeomorphisms and their total valencies}

    
    We first recall the classification of the conjugacy classes of periodic diffeomorphisms on $\Sigma_{g}$ in terms of the total valency introduced in \cite{Ashikaga-Ishizaka}.
	Let $f \in \operatorname{Diff}_+(\Sigma_{g})$ be a periodic diffeomorphism of order $n$ and $C$ the cyclic subgroup of $\operatorname{Diff}_{+}(\Sigma_{g})$ generated by $f$.
	An $C$-orbit $Cx$ is called {\it multiple} if $|Cx| < |C|$, where $|\cdot|$ denote the cardinality.
	For $x \in \Sigma_g$, the minimum positive integer $\ell$ such that $f^{\ell}(x)=x$ is called the period of $Cx$.
	Let $\lambda = |C_{x}|$, where $C_{x}$ is the isotropy group of $x$.
	Note that $\frac{n}{\lambda}$ is equal to the period of $Cx$.
	Then, there uniquely exists $\mu \in \{1,2, \ldots, \lambda - 1\}$ such that that the restriction of $f$ to a small neighborhood of $x$ is the clockwise $\frac{2\pi\mu}{\lambda}$-rotation.
	Since $\mu$ and $\lambda$ are coprime, there exists an integer $\theta \in \{1,2, \ldots, \lambda - 1\}$ such that $\mu \theta \equiv 1 \ \mod \lambda$.
	The {\it valency} of $Cx$ is defined by $\frac{\theta}{\lambda}$.
	Let $\frac{\theta_1}{\lambda_1}, \frac{\theta_2}{\lambda_2}, \ldots, \frac{\theta_s}{\lambda_s}$ be the valencies of all multiple orbits of $f$.
	The data $[ \, g,n \,; \frac{\theta_1}{\lambda_1} + \frac{\theta_2}{\lambda_2} + \cdots + \frac{\theta_s}{\lambda_s} \, ]$ is called the {\it total valency} of $f$.
		
		\begin{figure}[ht]
			\centering
			\includegraphics[width=0.5\textwidth]{Figures/Sig2.pdf}
			\caption{A loop around a vertex $y$}
			\label{fig:Sig2}
		\end{figure}
	
	By the following theorem, total valencies exactly determine periodic diffeomorphisms on $\Sigma_{g}$ up to conjugation.
	
		\begin{thm}[{Nielsen \cite[Section 11]{Nielsen}, see also \cite[Section 1.3]{Ashikaga-Ishizaka}, \cite[Theorem 2.1]{Hirose}}]
		\label{thm:Nielsen}
			Let $f, f'$ be periodic diffeomorphisms of $\Sigma_{g}$ with total valencies $[\,g,n \,;\, \frac{\theta_1}{\lambda_1} + \frac{\theta_2}{\lambda_2} + \cdots +\frac{\theta_s}{\lambda_s} \,]$ and $[\,g,n' \,;\,\frac{\theta_1'}{\lambda_1'} + \frac{\theta_2'}{\lambda_2'} + \cdots +\frac{\theta_{s'}'}{\lambda_{s'}'} \,]$, respectively.
			$f$ is conjugate to $f'$ if and only if the following are satisfied:
				\begin{enumerate}
				\renewcommand{\labelenumi}{(\roman{enumi})}
					\item $s=s'$,
					\item $n=n'$,
					\item after changing indices, we have $\frac{\theta_i}{\lambda_i}=\frac{\theta_i'}{\lambda_i'}$ for $i =1,2,\ldots,s$.
				\end{enumerate}
		\end{thm}
	
	By Theorem \ref{thm:Nielsen}, we can identify the total valency of a periodic diffeomorphism with its conjugacy class.
	We will use the following well-known observations of Nielsen.

		\begin{prop}[{Nielsen \cite[Equation(4.6)]{Nielsen}}]
		\label{prop:Nielsenint}
			The sum of valencies of all multiple orbits of a periodic diffeomorphism is an integer.
		\end{prop}
		
	    \begin{rem}
		\label{rem:inv}
        	Any multiple orbit of an involution is a fixed point whose valency is $\frac{1}{2}$.
			Thus, by Proposition \ref{prop:Nielsenint}, the number of fixed points of an involution is even.
	    \end{rem}

	In general, the composite of two periodic diffeomorphism may not be periodic, and hence, the product of total valency does not make sense.
	But one can define powers of total valency of a periodic diffeomorphism $f$ by the total valency of powers of $f$.

    Let us recall the well-known Riemann-Hurwitz formula for the convenience of the reader.
		
		\begin{prop}[{Riemann-Hurwitz formula}]
		\label{prop:RH}
			Consider the $n$-fold branched covering from $\Sigma_g$ to $\Sigma_{g'}$ with branched indices $\lambda_1, \ldots, \lambda_s$.
			Then we have the formula
				\[
					2g-2 = n \left( 2g' -2 + \displaystyle\sum_{1 \leq i \leq s} \left( 1 - \frac{1}{\lambda_i} \right) \right).
				\]
		\end{prop}

    We will use the following result due to Harvey, which gives a necessary condition for branched indices.
	
		\begin{prop}[{Harvey \cite[Theorem 4]{Harvey}}]
		\label{prop:Harvey}
		    Assume $g>1$.
			Set $M = \operatorname{lcm}(\lambda_1,\lambda_2,\cdots , \lambda_s)$.
				If there is cyclic covering from $\Sigma_g$ to $\Sigma_{g'}$ with branched indices $\lambda_1,\cdots, \lambda_s$, then the following conditions are satisfied:
					\begin{enumerate}
					\renewcommand{\labelenumi}{(\roman{enumi})}
						\item $\operatorname{lcm}(\lambda_1,\lambda_2,\cdots, \hat{\lambda_i}, \cdots , \lambda_s) = M$ for all $i \in \{ 1,2,\ldots, s\}$, where $\hat{\lambda_i}$ denotes the omission of $\lambda_i$.
						\item $M$ divides $n$, and if $g' = 0$, then $M=n$.
						\item $s \neq 1$, and, if $g'=0$, then $s \geq 3$.
					\end{enumerate}
		\end{prop}
		
	We will use the following characterization of irreducibility of finite groups actions due to Gilman.
		
		\begin{thm}[{Gilman \cite[Lemma 3.9]{Gilman}}]
		\label{thm:Gilman}
			Let $G$ be a finite subgroup of $\operatorname{Diff}_+(\Sigma_g)$.
			Then $G$ is irreducible if and only if $(\Sigma_g \backslash M_G) / G$ is homeomorphic to $S^2 \backslash \{3 \text{-points}\}$.
		\end{thm}
	
	
\section{Preliminaries}
    
	In this section, we will classify irreducible periodic diffeomorphisms which commutes with an involution $\iota_{g}$ such that $\Sigma_{g}/\iota_{g}$ is a torus to prove Theorem \ref{thm:irr1}.
	Firstly, let us compute the total valency of $h_{n,p}$ defined in the introduction.

		\begin{prop}
		\label{prop:val}
			The total valency of $h_{n,p}$ is $[\, g, n\, ; \, \frac{1}{n} + \frac{p}{n} + \frac{n-p-1}{n} \,]$, where $g=\frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}$.
		\end{prop}
		
		\begin{proof}
    		The genus $g$ of the surface can be computed by counting the edges of the $2n$-gon $P$ (see Figure \ref{fig:Sig}) as 
        		\[
    	        	g=\frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, p+1) + 1}{2}.
        		\]
			Clearly $h_{n,p}$ has $3$ multiple orbits: The barycenter of $P$, the vertices of $P$ and the middle point of the edges of $P$.
			Thus, the total valency of $h_{n, p}$ is of the form $[ \,  g, n \,;\, \frac{\theta_1}{\lambda_1}+ \frac{\theta_2}{\lambda_2}+ \frac{\theta_3}{\lambda_3} \, ]$ for some coprime integer $\lambda_i, \theta_i$ such that $\lambda_i$ divides $n$ for $i = 1, 2, 3$.
			
			It is clear that the valency of the fixed point at the barycenter of the $2n$-gon is $\frac{1}{n}$.
			Hence, we can set $\frac{\theta_1}{\lambda_1} = \frac{1}{n}$.
			
			Let us compute $\frac{\theta_2}{\lambda_2}$ the valency of the multiple orbit which is consisted by the body-centered points of the barycenter of Figure \ref{fig:Sig}.
			
				\begin{figure}[ht]
				    \centering
			        \captionsetup{width=0.6\linewidth}
			        \includegraphics[width=0.45\textwidth]{Figures/Sigval.pdf}
				    \caption{The cyclic order of\\ fundamental domains of $h_{n,p}$ around $y$}
				    \label{fig:Sigval}
		        \end{figure}
		
			Set $k = \opn{gcd} (n, p)$.
			Let $n'$ and $p'$ be the integer such that $n = k n'$ and $p = k p'$, respectively.
			The period of the multiple orbit is $k$, and hence, $\lambda_2 = n'$.
			Take a point $y$ in the orbit, and consider a loop which goes around the boundary of a small disk centered at $y$ as shown in Figure \ref{fig:Sigval}.
			The fundamental domains appear along this loop in the order
				\[
					0, \, p, \, 2p, \ldots,  (n'-2)p, \, (n'-1)p
				\]
			modulo $n$.
			Then, there uniquely exists $\nu \in \left\{ 1, 2, \ldots, n' \right\}$ such that $\nu p \equiv k \, \mod n$.
			Hence, we have $\nu p' \equiv 1\, \mod n'$.
			By the definition of a valency, $\theta_2 = p'$, and hence, $\frac{\theta_2}{\lambda_2} = \frac{p'}{n'} = \frac{p}{n}$.
			
			By Proposition \ref{prop:Nielsenint}, the valency $\frac{\theta_3}{\lambda_3}$ of another multiple orbit holds the condition
				\[
					\frac{1}{n} + \frac{p}{n} + \frac{\theta_3}{\lambda_3} \in \mathbb{Z}.
				\]
					Thus, we have $\frac{\theta_3}{\lambda_3} = \frac{n-p-1}{n}$.
		\end{proof}

    Applying the Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we can obtain the following well-known classification of periodic diffeomorphisms on tori.

        \begin{prop}
        \label{prop:brto}
            Let $f$ be a nontrivial orientation preserving periodic diffeomorphism on a torus $T$.
            Then $T/\langle f \rangle$ is homeomorphic to either $T^{2}$ or $S^{2}$.
            
            If $T/\langle f \rangle \cong T^2$, then $f$ is multiple orbit free.
            If $T/\langle f \rangle \cong S^2$, then the total valency of $f$ is one of the following:
                    \begin{align*}
                    h_{4,1}^2 = h_{6,3}^3 = & \left[\,1,2\,;\, \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}\,\right], && \\
                    h_{4,1} = & \left[\,1,4\,;\, \frac{1}{4} + \frac{1}{4} + \frac{1}{2}\,\right], &
                    h_{4,1}^{3}= & \left[\,1,4\,;\, \frac{3}{4} + \frac{3}{4} + \frac{1}{2}\,\right], \\
                    h_{6,3} = & \left[\,1,6\,;\, \frac{1}{6} + \frac{1}{3} + \frac{1}{2}\,\right], &
                    h_{6,3}^{2}= & \left[\,1,3\,;\, \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\,\right], \\
                    h_{6,3}^{4}= & \left[\,1,3\,;\, \frac{2}{3} + \frac{2}{3} + \frac{2}{3}\,\right], &
                    h_{6,3}^{5}= & \left[\,1,6\,;\, \frac{5}{6} + \frac{2}{3} + \frac{1}{2}\,\right].
                \end{align*}
        \end{prop}
        
    It is easy to see that we can identify $h_{3,1}$ with $h_{6,3}^2$ by deforming the $12$-gon to define $h_{6,3}$ to a hexagon (see Figures \ref{fig:1232}, see also Lemma \ref{lem:Inv}(1)).
    
        \begin{figure}[ht]
        \begin{tabular}{cc}
            \centering
            \begin{minipage}{0.45\textwidth}
                \captionsetup{width=0.85\linewidth}
                \includegraphics[width=1\textwidth]{Figures/1232.pdf}
                \caption{$h_{6,3}^2$}
                \label{fig:1232}
            \end{minipage}\hfill
            \begin{minipage}{0.45\textwidth}
                \captionsetup{width=0.85\linewidth}
                \includegraphics[width=0.9\textwidth]{Figures/111.pdf}
                \caption{$h_{3,1}$}
                \label{fig:h31}
            \end{minipage}
        \end{tabular}
        \end{figure}

    Since any multiple point free periodic diffeomorphism on a torus is reducible, we get the consequence.
    
        \begin{prop}
            Every irreducible periodic diffeomorphism on a torus is conjugate to one of the following: 
                \begin{enumerate}
                    \item $h_{4,1}$ or its inverse, 
                    \item $h_{6,3}$ or its inverse,
                    \item $h_{3,1}$ or its inverse.
                \end{enumerate}
        \end{prop}

	Here $h_{4,1}$, $h_{6,3}$ and $h_{3,1}$ are shown in Figures \ref{fig:112}, \ref{fig:123} and \ref{fig:h31}, respectively.

		\begin{figure}[ht]
			\centering
			\begin{tabular}{cc}
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/112.pdf}
				\caption{$h_{4,1}$}
				\label{fig:112}
			\end{minipage}\hfill
			\begin{minipage}{0.45\textwidth}
				\centering
				\includegraphics[width=1\textwidth]{Figures/12312.pdf}
				\caption{$h_{6,3}$}
				\label{fig:123}
			\end{minipage}
			\end{tabular}
		\end{figure}
		
\section{Cyclic branched coverings of tori}
    
	Take an involution $\iota_g$ of $\Sigma_g$ whose quotient space is homeomorphic to a torus.
	Let $f$ be a periodic diffeomorphism of $\Sigma_g$ of order $n$ which commutes with $\iota_g$.
	
	Let $\bar{f}$ be a periodic diffeomorphism of $\Sigma_{g} / \langle \iota_g \rangle$ induced by $f$.
	Let $n=\operatorname{ord}(f)$ and $\bar{n}=\operatorname{ord}(\bar{f})$.
	We have the following simple observation.

		\begin{lem}
		\label{lem:1}
			\begin{enumerate}
				\item We have $n = 2\bar{n}$ if and only if $f^{\bar{n}} = \iota_g$.
				\item We have $n = \bar{n}$ if and only if $f^{\bar{n}} \neq \iota_g$.
			\end{enumerate}
		\end{lem}
		
		\begin{proof}
			We have $\bar{f}^{\bar{n}} = \operatorname{id}_{\Sigma_{g}/\langle \iota_g \rangle}$.
			Since the lifts of $\operatorname{id}_{\Sigma_{g} / \langle \iota_g \rangle}$ to $\Sigma_{g}$ are $\iota_g$ and $\operatorname{id}_{\Sigma_{g}}$, we have either $f^{\bar{n}} = \iota_g$ or $f^{\bar{n}} = \operatorname{id}_{\Sigma_{g}}$.
			The latter is equivalent to $n = \bar{n}$, and the former implies $n = 2\bar{n}$.
			The proof is concluded.
		\end{proof}
		
	We will use the following well known result.
		
		\begin{lem}
        \label{lem:fix3}
            There exists no cyclic branched covering $p : S^{2} \to S^{2}$ which has more than two branched points $x_{1}, x_{2}, \dots, x_{s} \in S^{2}$ such that $p^{-1}(x_{i})$ is a one point set for $i=1,2,3$.
            In particular, there exists no nontrivial periodic diffeomorphism of an orbifold homeomorphic to $S^{2}$ which fixes more than two points.
        \end{lem}
        
        \begin{proof}
            Assume that there exists such an $n$-fold branched covering $p : S^{2} \to S^{2}$. 
            By Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we have
                \[
                    -2 = n \left( -2 + 3 \left( 1 - \frac{1}{n} \right) + \sum_{i=4}^{s}\left( 1 - \frac{1}{\lambda_{i}} \right) \right),
                \]
            where $\lambda_{i}$ is the ramification index of $x_{i}$ for $i=4, \dots, s$. Then we have 
                \[
                     1-n = \sum_{i=4}^{s}\left( 1 - \frac{1}{\lambda_{i}} \right) \geq 0,
                \]
            which implies $n \leq 1$, and hence a contradiction.
        \end{proof}
	
	Let $G$ be the group generated by $f$ and $\iota_g$. 
	In this section, we will classify the $G$-action up to conjugacy to prove Theorem \ref{thm:irr1}.
	
	The following simple observation is fundamental. 
	
		\begin{lem}
		\label{lem:pres}
		    \begin{enumerate}
		        \item We have $f \left(\operatorname{Fix} (\iota_{g}) \right) = \operatorname{Fix} (\iota_{g})$, where $\operatorname{Fix} (\iota_{g})$ is the fixed point set of $\iota_g$.
		        \item The involution $\iota_g$ maps every $f$-orbit to an $f$-orbit of the same valency.
		    \end{enumerate}
		\end{lem}
		
    	\begin{proof}
        	It is easy to see that the valency of an $f$-orbit of $x$ is determined by the isotropy action of $f^{k}$, where $k$ is the period of $x$.
        	The lemma follows from the conjugation invariance of the isotropy action.
	    \end{proof}
	
	It is easy to determine the conjugacy class of the $G$-action in the case where $G$ is cyclic. Let $\bar{f}$ be the diffeomorphism on $\Sigma_g / \langle \iota_g \rangle$ induced by $f$.

	Let $p$ denote the double covering $\Sigma_g \to \Sigma_g / \langle \iota_g \rangle$. In the sequel, we will use the following criterion.
	
		\begin{lem}
		\label{lem:brodd}
			If $p$ is ramified at a multiple point $x$ of $f$ of period $k$ such that $\frac{n}{k}$ is even, then $G$ is cyclic.
		\end{lem}
		
		\begin{proof}
		    By assumption, the isotropy group $G_{x}$ of $x$ is generated by $f^{k}$ and $\iota_{g}$.
		    Since $\frac{n}{k}$ is even, the subgroup of $G_{x}$ generated by $f^{k}$ is of even order.
		    Thus, it contains $f^{\frac{n}{2}}$, which is a $\pi$-rotation around $x$.
		    Therefore, it follows that $f^{\frac{n}{2}} = \iota_{g}$, which implies that
		    $G$ is cyclic.
		\end{proof}
	
	It is easy to see that if $\bar{f}$ is reducible then $G$ is reducible: 
	Indeed, if $\bar{f}$ preserves a system of isotopy classes of essential simple closed curves, then $G$ preserves its inverse image of $p$. 
	Hence, to classify irreducible group $G$, it is sufficient to consider the case where $\bar{f}$ is irreducible.

		\begin{figure}[ht]
			\centering
				\includegraphics[width=0.45\textwidth]{Figures/simplebr.pdf}
				\caption{A simple closed curve $\mathcal{C}$}
				\label{fig:simplebr}
		\end{figure}
		
		\begin{lem}
		\label{lem:freered}
    		Assume that $\bar{f}$ is conjugate to a power of $h_{n,p}$.	
	    	If the canonical projection $p : \Sigma_g \to \Sigma_g / \langle \iota_g \rangle$ is branched at a free orbit of $\bar{f}$, then $G$ is reducible.
		\end{lem}
		
		\begin{proof}
			Assume that $p$ is branched at a simple point $y$ of $\bar{f}$.
			Take an $\bar{f}$-invariant simple closed curve $\mathcal{C}$ of $\Sigma_g / \langle \iota_g \rangle$ so that it bounds a disk which contains the $\bar{f}$-orbit of $y$ as shown in Figure \ref{fig:simplebr}.
			Then the preimage $p^{-1}(\mathcal{C})$ is essential and preserved by $f$ and $\iota_g$, hence $G$ is reducible.
		\end{proof}
	
	Now, we will determine irreducible $f$ that induces $h_{4,1}$ on the torus $\Sigma_{g} / \langle \iota_g \rangle$.

	Since $\iota_g$ has $2g-2$ fixed points, $p$ is branched at $2g-2$ points.
	By Lemma \ref{lem:pres}, $\bar{f}$ maps the branched locus of $p$ to itself.
	By classifying nonempty $\bar{f}$-invariant sets with an even number of points in the union of the multiple orbits of $\bar{f}$, we obtain the following lemma.
	
		\begin{lem}
		\label{lem:4}
			If $\bar{f}=h_{4,1}$, then the branched locus $B_{p}$ of $p$ is one of the following:
				\begin{enumerate}
					\item $\{ x, y \}$,\label{case:31}
					\item $\{ z_1, z_2 \}$,\label{case:32}
					\item $\{ x, y, z_1, z_2 \}$,\label{case:33}
				\end{enumerate}
			where $x, y, z_1, z_2 \in \Sigma_{g}/\langle \iota_g \rangle$ are as shown in Figure \ref{fig:112}.
		\end{lem}
		
		\begin{prop}
		\label{prop:1}
			If $\bar{f}=h_{4,1}$, then $G$ is conjugate to either of $\langle h_{8,1} \rangle$ or $\langle h_{8,5} \rangle$.
		\end{prop}
		
		\begin{proof}
		    We can see that $G$ is cyclic in all three cases in Lemma \ref{lem:4}.
		    Indeed, since $\operatorname{ord} (\bar{f}) = 4$, and the period of branch points is either $1$ or $2$, we can see that the cardinality of the isotropy group of every branch point is even.
		    Then, by Lemma \ref{lem:brodd}, $G$ is cyclic.
		    By Lemma \ref{lem:1}, we have that $n=8$. 
		
			Let us consider the case \ref{case:31} in Lemma \ref{lem:4}.
			We can see that $p$ has three branched points of index $8,8$ and $2$, respectively.
			Then the total valency of $f$ is of the form $[\,2,8\, ; \, \frac{\theta_1}{8}+ \frac{\theta_2}{8}+ \frac{1}{2} \,]$, where $\theta_1, \theta_2 \in \{ 1,3,5,7 \}$.
			Here $\frac{\theta_1}{8} + \frac{\theta_2}{8} + \frac{1}{2}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}), and hence, $(\theta_1, \theta_2) = (1,3)$ or $(5,7)$.
			In both cases, since $f^4$ is a hyperelliptic involution, it contradicts Lemma \ref{lem:1}.
			Thus, this case does not occur.
		
			In the case \ref{case:32} of Lemma \ref{lem:4}, we can see that $p$ has three branched points of index $4$.
			Thus the total valency of $f$ is of the form $[\,2,8\, ; \, \frac{\theta_1}{4}+ \frac{\theta_2}{4}+ \frac{\theta_3}{4} \,]$, where $\theta_{1}, \theta_{2}, \theta_{3} \in \{1,3\}$.
			It is clear that there are no $\theta_{1}, \theta_{2}, \theta_{3}$ such that $\frac{\theta_1}{4}+ \frac{\theta_2}{4}+ \frac{\theta_3}{4}$ is an integer.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), this case does not occur.
		
			Finally, let us consider the case \ref{case:33} of Lemma \ref{lem:4}.
			We can see that $p$ has three branched points of index $8,8$ and $4$, respectively.
			Let $[\,2,8\, ; \, \frac{\theta_1}{8}+ \frac{\theta_2}{8}+ \frac{\theta_3}{4} \,]$ be the total valency of $f$, where $\theta_1, \theta_2 \in \{ 1,3,5,7 \}$ and $\theta_3 \in \{1,3\}$.
			Now $\frac{\theta_1}{8} + \frac{\theta_2}{8} + \frac{\theta_3}{4}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}), and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,3k)$ or $(k,5k,k) \mod 8$ for $k \in \{ 1,3,5,7 \}$.
			Thus, $f$ is a power of either of $h_{8,1}=[\, 3,8 \,;\, \frac{1}{8}+\frac{1}{8}+\frac{3}{4} \,]$ or $h_{8,5}=[\, 3,8 \,;\, \frac{1}{8}+\frac{5}{8} +\frac{1}{4} \,]$.
		\end{proof}

    Let us consider the case where $\bar{f} = h_{6,3}$.
    We obtain the following lemma by an argument similar to the case where $\bar{f} = h_{4,1}$.

		\begin{lem}
		\label{lem:5}
			If $\bar{f}=h_{6,3}$, then branched locus $B_p$ of $p$ is one of the following:
				\begin{enumerate}
					\item $\{ y_1, y_2 \}$,\label{case:41}
					\item $\{ x, z_1, z_2, z_3 \}$,\label{case:42}
					\item $\{ x, y_1, y_2, z_1, z_2, z_3 \}$,\label{case:43}
				\end{enumerate}
			where $x, y_1, y_2, z_1, z_2, z_3 \in \Sigma/ \langle \iota_g \rangle$ are as shown in Figure \ref{fig:123}.
		\end{lem}
		
		\begin{prop}
		\label{prop:2}
			If $\bar{f}=h_{6,3}$, then $G$ is conjugate to a subgroup of either of $\langle h_{6,1}, I \rangle, \langle h_{12,2} \rangle$, or $\langle h_{12,3} \rangle$.
		\end{prop}
		
		\begin{proof}
		 	Let us consider the case \ref{case:42} of Lemma \ref{lem:5}.
		 	Since $p$ is branched at a fixed point $x$ of $\bar{f}$ and the order of $f$ is even, by Lemma \ref{lem:brodd}, $G$ is cyclic.
			Then, by Lemma \ref{lem:1}, we have $n=12$ and the branch indices of $f$ are $12,4$ and $3$.
			Let $[\,3,12\, ; \, \frac{\theta_1}{12}+ \frac{\theta_2}{4}+ \frac{\theta_{3}}{3} \,]$ be the total valency of $f$, where $\theta_1 \in \{ 1,5,7,11 \}, \theta_2 \in \{ 1,3 \}$ and $\theta_3 \in \{1,2 \}$.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta_1}{12} + \frac{\theta_2}{4} + \frac{\theta_3}{3}$ is an integer, and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,2k) \mod 12$ for $k \in \{ 1,5,7,11 \}$.
			Thus, $f$ is a power of $h_{12,3}=[\, 3,12 \,;\, \frac{1}{12}+\frac{1}{4}+\frac{2}{3} \,]$ shown in Figure \ref{fig:138}.
			Note that $h_{12,3}^6$ is equal to $\iota_3$.
			
			Let us consider the case \ref{case:43} of Lemma \ref{lem:5}.
			Since $p$ is branched at a fixed point $x$ of $\bar{f}$ and the order of $f$ is even, by Lemma \ref{lem:brodd}, $G$ is cyclic.
			Then, by Lemma \ref{lem:1}, we have $n=12$, and the branch indices of $f$ are $12,6$ and $4$.
			Let $[\,3,12\, ; \, \frac{\theta_1}{12}+ \frac{\theta_2}{6}+ \frac{\theta_{3}}{4} \,]$ be the total valency of $f$, where $\theta_1 \in \{ 1,5,7,11 \}, \theta_2 \in \{ 1,5 \}$ and $\theta_3 \in \{1,3 \}$.
			By Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta_1}{12} + \frac{\theta_2}{6} + \frac{\theta_3}{4}$ is an integer, and hence, $(\theta_1, \theta_2,\theta_3) = (k,k,3k) \mod 12$ for $k \in \{ 1,5,7,11 \}$.
			Thus, $f$ is a power of $h_{12,2}=[\, 3,12 \,;\, \frac{1}{12}+\frac{1}{6}+\frac{3}{4} \,]$.
			Note that $h_{12,2}^{6}$ is equal to $\iota_4$.
			
			Finally, let us consider the case \ref{case:41} in Lemma \ref{lem:5}.
	    	Since $p$ is branched at $2$ points, we have $g=2$.
			Since $\bar{f}$ fixes $x$, it follows that $p^{-1}(x)$ is either a multiple orbit of period two, or $f$ fixes $p^{-1}(x)$ pointwisely.
			In the former case, $\iota_{g} \circ f$ is a lift of $\bar{f}$ which fixes $p^{-1}(x)$ pointwisely.
			Thus, to classify the conjugacy class of $G$, we can assume that $f$ fixes $p^{-1}(x)$ pointwisely without loss of generality.
			In this case, we can see that $n=6$.
			Since the valency of $\{y_{1},y_{2}\}$ as an $\bar{f}$-orbit is $\frac{1}{3}$, by a local argument, we can see that the valency of $p^{-1}(\{y_1, y_2 \})$ is either $\frac{2}{3}$ or $\frac{1}{6}$.
			But $n=6$ yields that only the former case occurs.
			Similarly, $p^{-1}(\{ z_1, z_2, z_3 \})$ is either a free orbit or the union of two multiple orbits of valency $\frac{1}{2}$.
		 	The Riemann-Hurwitz formula (Proposition \ref{prop:RH}) yields that only the former case occurs here.
		 	Indeed, if we substitute $g=2$ and $(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}) = (6,6,3,2,2)$ to the formula,
			then we get $g' = -\frac{1}{2}$, which contradicts to the fact that $g'$ is the genus of $\Sigma_{2}/\langle f \rangle$.
			Thus, it follows that the total valency of $f$ is $[\,2,6\,;\, \frac{1}{6}+\frac{1}{6}+\frac{2}{3} \,]$.
			Thus $f$ is conjugate to $h_{6,1}$.
			Now $f^{3} \circ \iota_{2}$ is a hyperelliptic involution which fixes all points in $p^{-1}(\{ z_1, z_2, z_3 \})$.
			By \cite[Lemma 13]{TN}, $G$ is conjugate to $\langle h_{6,1}, I \rangle$, where $I$ is a hyperelliptic involution on $\Sigma_{2}$.
		\end{proof}

    Finally, let us consider the case where $\bar{f} = h_{3,1}$.
    The multiple orbits of $h_{3,1}$ are three fixed points.
    A nonempty $\bar{f}$-invariant subset with an even number of points in the set of these three fixed points is a two point set.
    All of them are mutually conjugate, and hence we can assume that $p$ is branched at $y$ and $z$ in Figure \ref{fig:h31}.
    Thus, the branch locus of $p$ is the same as the case the case \ref{case:41} in Lemma \ref{lem:5}.
    By a parallel argument, we can see that $G$ is conjugate to a subgroup of $\langle h_{6,1}, I \rangle$.

    Theorem \ref{thm:irr1} follows from Propositions \ref{prop:1}, \ref{prop:2} and the argument in the last paragraph.

\section{Finite subgroups of the centralizer of irreducible periodic diffeomorphisms on surfaces}


    In this chapter, we will classify up to conjugacy finite subgroups of the centralizer of irreducible periodic diffeomorphisms to prove Theorem \ref{thm:irr2}.

    Let $f$ be an irreducible periodic diffeomorphism of $\Sigma_g$ of period $n$.
	By Theorem \ref{thm:Gilman}, the total valency of $f$ is of the form $[\,g,n\,;\, \frac{\theta_1}{n} + \frac{\theta_2}{n} + \frac{\theta_3}{n} \,]$ for some $\theta_{1}$, $\theta_{2}$, $\theta_{3}$ such that $1 \leq \theta_{i} \leq \frac{n}{2}$ for $i=1,2,3$.
	Let $F$ be the subgroup of $\operatorname{Diff}_+ (\Sigma_g)$ generated by $f$, and let $H$ be a finite subgroup in the centralizer of $f$ in $\operatorname{Diff}_+ (\Sigma_g)$.

    	\begin{lem}
		\label{lem:diff}
			If $\theta_1, \theta_2$ and $\theta_3$ are mutually different, then $H$ is a subgroup of $F$.
		\end{lem}
		
		\begin{proof}
	    	Take an arbitrary $d \in H$.
	    	The diffeomorphism $d$ maps a multiple orbit of $f$ to another multiple orbit of the same valency by Lemma \ref{lem:pres}.
	    	Therefore, by assumption, $d$ maps each multiple orbit of $f$ to itself.
	    	Then $d$ induces $\bar{d}$ on $\Sigma/ F \cong S^2$, which fixes $3$ cone points.
	    	By Lemma \ref{lem:fix3}, it follows that $\bar{d}=\operatorname{id}_{\Sigma_g / F}$.
			Since any lift of $\operatorname{id}_{\Sigma_g / F}$ is of the form $f^k$ for some integer $k$, we have $d=f^{k}$.
			Therefore $H$ is a subgroup of $F$.
		\end{proof}
	
	Now let us consider the case where $\theta_{1} = \theta_{2}$.
	We observe the following.
	
	    \begin{lem}\label{lem:g1}
	        If $\theta_{1} = \theta_{2} = \theta_{3}$, then $f$ is conjugate to $h_{3,1}$ or $h_{3,1}^{2}$.
	        In particular, we have $g=1$.
        \end{lem}

        \begin{proof}
            Let $\theta=\theta_{1}=\theta_{2}=\theta_{3}$.
            Here $\frac{3\theta}{n}$ is an integer by Nielsen's condition (Proposition \ref{prop:Nielsenint}).
            Since $0 < \frac{\theta}{n} < 1$, we have $\frac{3\theta}{n}=1$ or $2$.
            In the case where $\frac{3\theta}{n}=1$, we have $(n,\theta) = (3,1)$.
            Otherwise we have $\frac{3\theta}{n}=2$, and hence $(n,\theta) = (3,2)$.
            In both cases, by the Riemann-Hurwitz formula (Proposition \ref{prop:RH}), we have $g=1$. Thus the total valency of $f$ is equal to either $[\, 1,3\,;\frac{1}{3}+\frac{1}{3}+\frac{1}{3} \,]$ or $[\,1,3\,;\frac{2}{3}+\frac{2}{3}+\frac{2}{3}\,]$.
            The former is the total valency of $h_{3,1}$, and the latter is the total valency of $h_{3,1}^{2}$.
            Thus, $f$ is conjugate to $h_{3,1}$ or $h_{3,1}^{2}$ by Theorem \ref{thm:Nielsen}.
        \end{proof}

    Thus, in order to prove Theorem \ref{thm:irr2}, it is sufficient to consider the case where
        \[
            \theta_1=\theta_2\neq \theta_3.
        \]
    By the first condition of Proposition \ref{prop:Harvey}, $\operatorname{gcd}(\theta_1,n)=1$, and hence, $f$ is conjugate to the $\theta_1$-th power of $h_{n,1}$.
    Clearly $H$ is conjugate to a finite subgroup of the centralizer of $h_{n,1}$. Thus, we can assume that $f=h_{n,1}$ and $H$ is a finite subgroup of the centralizer of $h_{n,1}$ in the following. Since $g = \frac{n-\operatorname{gcd}(n, p) - \operatorname{gcd}(n, n-p-1) + 1}{2}$, we can compute the period $n$ in terms of the genus $g$ as follows:
        \[
            n = \begin{cases}
                    2g+1 & n : \text{odd,} \\
                    2g+2 & n : \text{even.}
                 \end{cases}
        \]
    
    Since $[h_{n,1}]=[\,g,n\,;\,\frac{1}{n}+\frac{1}{n}+\frac{n-2}{n}\,]$, the diffeomorphism $h_{n,1}$ have two fixed points of valency $\frac{1}{n}$ and a multiple orbit of valency $\frac{n-2}{n}$, which is a fixed point if $n$ is odd and has period $2$ if $n$ is even.
    We have, for every $n$, a (hyperelliptic) involution which commutes with $h_{n,1}$ and exchanges two fixed points of valency $\frac{1}{n}$ as follows:
    
        \begin{lem}
        \label{lem:Inv}
            \begin{enumerate}
                \item[ ] 
                \item $h_{4g+2,2g+1}^{2g}$ is conjugate to $h_{2g+1,1}$, and $h_{4g+2,2g+1}^{2g+1}$ is a hyperelliptic involution which commutes with $h_{4g+2,2g+1}^{2g}$ and exchanges two fixed points of $h_{4g+2,2g+1}^{2g}$ of valency $\frac{1}{2g+1}$.
                \item $I$ shown in Figure \ref{fig:Iphi3} is a hyperelliptic involution which commutes with $h_{2g+2,1}$ and exchanges two fixed points of $h_{2g+2,1}$ of valency $\frac{1}{2g+2}$.
           \end{enumerate}
        \end{lem}

        \begin{proof}
            By Proposition \ref{prop:val}, the total valency of $h_{4g+2,2g+1}$ is equal to $[\,g,4g+2\,;\, \frac{1}{4g+2} + \frac{g}{2g+1} + \frac{1}{2} \,]$.
            Since $\operatorname{gcd} (4g+2, 2g)=2$, the period of $h_{4g+2,2g+1}^{2g}$ is $2g+1$.
            Figure \ref{fig:h4g+2} is a decomposition of $\Sigma_g$ into the fundamental domain of $h_{4g+2,2g+1}$.
            
            The multiple orbit of $h_{4g+2,2g+1}$ with valency $\frac{1}{2}$ consists of $2g+1$ points, and it is a free orbit of $h_{4g+2,2g+1}^{2g}$.
            
            By construction, $h_{4g+2,2g+1}$ is induced from the clockwise $\frac{2\pi}{4g+2}$-rotation the $8g+4$-gon.
            Thus, $h_{4g+2,2g+1}^{2g}$ is induced from $\frac{2g\pi}{2g+1}$-rotation (See Figure \ref{fig:h2g+1}).
            Let $x$ be the barycenter of Figure \ref{fig:h4g+2}.
            Since we have
                \[
                    g (2g-1) \equiv -2g \equiv 1 \ \mod 2g+1,
                \]
            by the definition of valency, the valency of $x$ is equal to $\frac{2g-1}{2g+1}$.
            
                \begin{figure}[ht]
	        	\centering
	        	\begin{tabular}{cc}
	        	\begin{minipage}{0.45\textwidth}
			    	\centering
	        		\captionsetup{width=0.87\linewidth}
		            \includegraphics[width=0.94\textwidth]{Figures/h4g+2.pdf}
    				\caption{Fundamental domains of $h_{4g+2}$}
	    			\label{fig:h4g+2}
		    	\end{minipage}\hfill
			    \begin{minipage}{0.45\textwidth}
				    \centering
		    		\captionsetup{width=0.87\linewidth}
			    	\includegraphics[width=0.94\textwidth]{Figures/h2g+1.pdf}
				    \caption{Fundamental domains of $h_{4g+2,2g+1}^{2g}$}
			    	\label{fig:h2g+1}
			    \end{minipage}
		    	\end{tabular}
            	\end{figure}
            
            The period of the other multiple orbit of $h_{4g+2,2g+1}$ is $2$.
            Thus $h_{4g+2,2g+1}^{2g}$ fixes this orbit pointwisely.
            Let $\frac{\theta}{2g+1}$ be its valency.
            Then, by Nielsen's condition (Proposition \ref{prop:Nielsenint}), $\frac{\theta}{2g+1}+\frac{\theta}{2g+1}+\frac{2g-1}{2g+1}$ is an integer, and hence, we have $\theta=1$.
            Therefore, we obtain 
                \[
                    [h_{4g+2,2g+1}^{2g}] = \left[\,g,2g+1\,\,;\,\frac{1}{2g+1}+\frac{1}{2g+1}+\frac{2g-1}{2g+1}\,\right] = [h_{2g+1,1}],
                \]
            which concludes the proof by Theorem \ref{thm:Nielsen}.
        \end{proof}
        
        \begin{lem}
            \begin{enumerate}
                \item $H$ is conjugate to a subgroup of $\langle h_{4g+2,2g+1} \rangle$, where $h_{2g+1,1}$ is identified with $h_{4g+2,2g+1}^{2g}$.
                \item $H$ is conjugate to a subgroup of $\langle h_{2g+2,1}, I\rangle$.
            \end{enumerate}
        \end{lem}

        \begin{proof}
            By Lemma \ref{lem:Inv}, for every $n$, we have an involution $I_{n}$ which commutes with $h_{n,1}$ and two fixed points of valency $\frac{1}{n}$.
            By Lemma \ref{lem:pres}, one of the following two cases occurs:
			
			    \begin{enumerate}
			        \item[(A)] $d$ maps each of three multiple orbits of $h_{n,1}$ to itself.
			        \item[(B)] $d$ exchanges two fixed points of $h_{n,1}$ with valency $\frac{1}{n}$, and maps the multiple orbit of $f$ with valency $\frac{n-2}{n}$ to itself.
			    \end{enumerate}

            It is easy to see that, by a similar argument to Lemma \ref{lem:diff}, we have $d \in F$ in the case (A).
            Thus, it is sufficient to consider the case where there exists $d \in H$ which satisfies the condition (B).
            Note that, for any $d' \in H$ which satisfies the condition (B),
            $d^{-1} \circ d'$ maps every three multiple orbits to itself.
            Then, Lemma \ref{lem:fix3} yields that $d^{-1} \circ d' = h_{n,1}^{k}$ for some $k$.
            Therefore $d' \in \langle h_{n,1}, d\rangle$.
            
            Consider the map $\bar{d}$ induced on $\Sigma_{g}/F$ by $d$.
            The map $I_{n}$ also induces a diffeomorphism $\bar{I}_{n}$ on $\Sigma_{g}/F$.
            Since both $\bar{d}$ and $\bar{I}_{n}$ are involutions which fix the same cone point,
            we have an orbifold diffeomorphism $\kappa : (\Sigma_{g}/F)/\bar{d} \to (\Sigma_{g}/F)/\bar{I}$.
            Then, by lifting $\kappa$ to a map $\Sigma_{g}/F \to \Sigma_{g}/F$, we have an orbifold diffeomorphism $\kappa_{1}$ which makes the following diagram commutative:
                \[
                    \xymatrix{
                    \Sigma_{g}/F \ar[r]^{\kappa_{1}} \ar[d] & \Sigma_{g}/F \ar[d] \\
                    (\Sigma_{g}/F)/\bar{d} \ar[r]_{\kappa} & (\Sigma_{g}/F)/\bar{I},
                    }
                \]
            where the vertical maps are the canonical projections.
            Then, since $\kappa$ is an isomorphism of double branched coverings, we have $\bar{d} \circ \kappa = \kappa \circ \bar{I}_{n}$.
            Therefore $\bar{d}$ is conjugate to $\bar{I}_{n}$.
            Then we may assume that $\bar{d} = \bar{I}_{n}$.
            Since $\bar{d} \circ \bar{I}_{n}^{-1} = \operatorname{id}_{\Sigma_{g}/F}$, we can see that $d \circ I_{n} \in F$.
            It follows that $d \in \langle h_{n,1}, I_{n} \rangle$.
        \end{proof}

    Theorem \ref{thm:irr2} follows from the last lemma and the preceding argument.


    We can see an exceptional phenomenon in the case of $g=1$.
    In this case, $f$ may be conjugate to $h_{3,1}$ or $h_{3,1}^{2}$ by Lemma \ref{lem:g1}.
    Since the centralizer of $f$ coincides up to conjugacy in these two cases, we can assume that $f=h_{3,1}$. As explained in the paragraph after Proposition \ref{prop:brto} (see Figures \ref{fig:1232}, \ref{fig:h31}), we can identify $h_{3,1}$ with $h_{6,3}^{2}$. Therefore $h_{6,3}$ commutes with $h_{3,1}$.
    In addition to $h_{6,3}$, we have one more periodic diffeomorphism $J$ which commutes with $h_{3,1}$ as shown in Figure \ref{fig:111J}.
    This $J$ and $h_{6,3}$ generates a finite group in the centralizer of $h_{3,1}$.

    \begin{figure}[ht]
	        \begin{tabular}{cc}
	            \begin{minipage}{0.44\textwidth}
	                \centering
	    		    \captionsetup{width=0.85\linewidth}
		        	\includegraphics[width=0.85\textwidth]{Figures/1111.pdf}
	            \end{minipage}\hfill \hspace{5pt}$\longrightarrow$
	       \hspace{2pt}
	       \begin{minipage}{0.44\textwidth}
	                \centering
	    		    \captionsetup{width=0.8\linewidth}
		        	\includegraphics[width=0.85\textwidth]{Figures/1112.pdf}
	            \end{minipage}
	        \end{tabular}
	        \caption{A periodic diffeomorphism $J$ which commutes with $h_{3,1}$}
	        \label{fig:111J}
	    \end{figure}
    
\input{references}
\end{document}
