  Let G be a finite group and let M be a G-manifold. We introduce the concept
of generalized orbifold invariants of M/G associated to an arbitrary group
Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected
manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a
natural and simple geometric origin in the context of locally constant
G-equivariant maps from G-principal bundles over covering spaces of Sigma to
the G-manifold M. We calculate generating functions of orbifold Euler
characteristic of symmetric products of orbifolds associated to arbitrary
surface groups (orientable or non-orientable, compact or non-compact), in both
an exponential form and in an infinite product form. Geometrically, each factor
of this infinite product corresponds to an isomorphism class of a connected
covering space of a manifold Sigma. The essential ingredient for the
calculation is a structure theorem of the centralizer of homomorphisms into
wreath products described in terms of automorphism groups of Gamma-equivariant
G-principal bundles over finite Gamma-sets. As corollaries, we obtain many
identities in combinatorial group theory. As a byproduct, we prove a simple
formula which calculates the number of conjugacy classes of subgroups of given
index in any group. Our investigation is motivated by orbifold conformal field
theory.
