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Let G be a finite group and let M be a G-manifold. We introduce the conceptof generalized orbifold invariants of M/G associated to an arbitrary groupGamma, an arbitrary Gamma-set, and an arbitrary covering space of a connectedmanifold Sigma whose fundamental group is Gamma. Our orbifold invariants have anatural and simple geometric origin in the context of locally constantG-equivariant maps from G-principal bundles over covering spaces of Sigma tothe G-manifold M. We calculate generating functions of orbifold Eulercharacteristic of symmetric products of orbifolds associated to arbitrarysurface groups (orientable or non-orientable, compact or non-compact), in bothan exponential form and in an infinite product form. Geometrically, each factorof this infinite product corresponds to an isomorphism class of a connectedcovering space of a manifold Sigma. The essential ingredient for thecalculation is a structure theorem of the centralizer of homomorphisms intowreath products described in terms of automorphism groups of Gamma-equivariantG-principal bundles over finite Gamma-sets. As corollaries, we obtain manyidentities in combinatorial group theory. As a byproduct, we prove a simpleformula which calculates the number of conjugacy classes of subgroups of givenindex in any group. Our investigation is motivated by orbifold conformal fieldtheory.

Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces