We consider conformal actions of the finite group G on a closed Riemann surface S, as well as algebraic actions of G on smooth, complete, algebraic curves over an arbitrary, algebraically closed field. There are several notions of equivalence of actions, the most studied of which is topological equivalence, because of its close relationship to the branch locus of moduli space. A second important equivalence relation is that induced by representation of G on spaces of holomorphic q-differentials. The notion of topological equivalence does not work well in positive characteristic. We shall discuss an alternative to topological equivalence, which we dub equisymmetry, that may be applied in all characteristics. The relation is induced by families of curves with G-action, and it works well with rotation constants and q-differentials, which are also defined in positive characteristic. After giving an overview of the various equivalence relations (conformal/algebraic, topological, q-differentials, rotation constants, equisymmetry) we focus on the interconnections among rotation constants, q-differentials, and equisymmetry.
Broughton, Sean A., "Topological and H^q Equivalence of Cyclic n-gonal Actions on Riemann Surfaces - Part II" (2020). Mathematical Sciences Technical Reports (MSTR). 175.