We introduce the –Euler–Satake
characteristics of a general orbifold
presented by an orbifold groupoid ,
extending to orbifolds that are not global quotients the generalized orbifold
Euler characteristics of Bryan–Fulman and Tamanoi. Each of these Euler
characteristics is defined as the Euler–Satake characteristic of the space of
–sectors of the
orbifold where
is a finitely generated discrete group. We study the behavior of these
Euler characteristics under product operations applied to the group
as
well as the orbifold and establish their relationships to existing Euler characteristics
for orbifolds. As applications, we generalize formulas of Tamanoi, Wang and Zhou for
the Euler characteristics and Hodge numbers of wreath symmetric products of
global quotient orbifolds to the case of quotients by compact, connected
Lie groups acting locally freely, in particular including all closed, effective
orbifolds.