Spaces of dihedral covers of curves and their irreducible components

Given a finite group G, consider the locus M_g(G), in M_g, consisting of curves which
admit an effective action by G. We propose numerical invariants of the G-action to
distinguish irreducible components of M_g(G). These invariants take into account the
local monodromies at the branch points of the associated Galois cover and certain
classes in H_2(G,Z), and extend those already introduced to study the case of abelian
groups.
When G=D_n, the dihedral group of order 2n, we show that these invariants
are in one-to-one correspondence with the irreducible components of M_g(D_n).
For general groups, we expect a bijection between irreducible components and
numerical invariants only after stabilization, when the genus g tends to infinity.
This is a joint work (in progress) with Fabrizio Catanese and Michael Loenne.