| Given a finite group G we study the locus M_g(G) (in the moduli space) of curves that admit an effective action by G. The topological type of the G-action provides a natural stratification of M_g(G) and a basic problem consists in defining numerical and homological invariants of G-actions that distinguish different topological types. We define a homological invariant for any G-action on a curve, extending the class in H_2(G,Z) usually associated to any free G-action. Moreover we show that, for any finite group G, this invariant is a fine and complete invariant for topological types of G-actions on curves of genus g, when g is sufficiently large. This extends a previous result of Dunfield-Thurston about the existence of a bijective correspondence between topological types of free G-actions on curves of genus g and H_2(G,Z)/Aut(G), for g large enough. We also report on works in progress aimed to understanding when two different topological types give rise to the same locus in the moduli space of curves. Seminar based on joint works with Fabrizio Catanese and Michael Lönne. |