Generalized orbifold Euler characteristics for general orbifolds and wreath products

We introduce the $\Gamma$-Euler-Satake characteristics of a general orbifold $Q$ presented by an orbifold groupoid $\mathcal{G}$, generalizing to orbifolds that are not necessarily 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 $\Gamma$-sectors of the orbifold where $\Gamma$ is a finitely generated discrete group. We study the behavior of these characteristics under product operations applied to the group $\Gamma$ 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 almost freely.