Genus stabilization for the homology of moduli spaces of orbit-framed curves with symmetries-I

In a previous paper, arXiv:1301.4409, we showed that the moduli space of curves C with a G-symmetry (that is, with a faithful action of a finite group G), having a fixed generalized homological invariant, is irreducible if the genus g' of the quotient curve C' : = C/G satisfies g'>>0. Interpreting this result as stabilization for the 0-th homology group of the moduli space of curves with G-symmetry, we begin here a program for showing stabilization for all the homology groups of these spaces, in similarity to the results of Harer for the moduli space of curves. In this first paper we prove homology stabilization for a variant of the moduli space where one G-orbit is tangentially framed, settling in particular homology stabilization for the case of at most one branch point.