Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.
Broughton, Sean A., "Topological and Hq Equivalence of Prime Cyclic p-gonal Actions on Riemann Surfaces" (2016). Mathematical Sciences Technical Reports (MSTR). 155.