Tamanoi Equation for Orbifold Euler Characteristics Revisited

Tamanoi equation is a Macdonald-type equation for the orbifold Euler characteristic and for its higher order analogs. It states that the generating series of fixed-order orbifold Euler characteristics of analogs of the symmetric powers for a space with a finite group action can be represented as a certain unified (explicitly written) power series raised to the power equal to the orbifold Euler characteristic of the same order of the space itself. In the paper, in particular, we explain how the Tamanoi equation follows from its verification for actions of (finite) groups on the one-point space. We generalize the statements used for this purpose to analogs of the orbifold Euler characteristic corresponding to finitely generated groups. We show that, for these generalizations, an analog of the Tamanoi equation does not hold in general.