if there exists an orientation preserving diffeomorphism $h$ of $S$ such that $h \phi (g)h^{-1}=\phi\sigma(g) $ for all $g\in G$. We denote a pair as above simply by $(C,G)$ (which is admittedly somewhat of an abuse of notation, as it hides the datum of the ${\operatorname{In}}({\operatorname{Mod}}(S))$-orbit of $\phi$) and we will often refer to it as a *$G$-curve* of topological type $\phi$.

The $G$-curves of topological type $\phi$ are parameterized by a D-M stack ${{\mathscr M}}^\phi$, which is defined as follows. Recall that the space of isotopy classes of complex structures on $S$ compatible with the given orientation is parameterized by a connected complex manifold, the Teichmüller space ${\operatorname{Teich}}(S)$, which supports a holomorphic family of curves ${{\mathscr C}}_{{\operatorname{Teich}}(S)}/{\operatorname{Teich}}(S)$. The mapping class group ${\operatorname{Mod}}(S)$ acts on this family. We let $G$ act on ${\operatorname{Teich}}(S)$ via $\phi$ (or what amounts to the same, via the natural map $G\to {\operatorname{Mod}}(S)$) and then its fixed point locus ${\operatorname{Teich}}(S)^G$ ([^2]) parameterizes the curves of topological type $\phi$. Since the $G$-invariant complex structures on $S$ are in bijective correspondence with the complex structures on $S_G$, we may identify ${\operatorname{Teich}}(S)^G$ with the Teichmüller space ${\operatorname{Teich}}(S_G)$ of the orbifold $S_G$.

The centralizer of $G$ in ${\operatorname{Mod}}(S)$, ${\operatorname{Mod}}(S)^G$, meets 