Let be a finite
group and let
be a –manifold.
We introduce the concept of generalized orbifold invariants of
associated to an
arbitrary group ,
an arbitrary –set,
and an arbitrary covering space of a connected manifold
whose fundamental
group is . Our
orbifold invariants have a natural and simple geometric origin in the context of locally constant
–equivariant maps from
–principal bundles
over covering spaces of
to the –manifold
. 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
. 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
–equivariant
–principal bundles
over finite –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.
