Genus stabilization for the components of moduli spaces of curves with symmetries

Abstract

In a previous paper (Groups, Geometry, and Dynamics, 2015), we introduced a new homological invariant for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant , if the genus g of the quotient curve satisfies g 0, is irreducible (and non-empty if and only if the class satisfies the 'admissibility' condition). We achieve this by showing that the stable equivalence classes of Hurwitz generating systems are in bijection with the admissible classes .