Orbifold Euler characteristics of non?orbifold groupoids

For a finitely presented discrete group $\Gamma$, we introduce two generalizations of the orbifold Euler characteristic and $\Gamma$-orbifold to class proper topological groupoids large enough include all cocompact Lie groupoids. The $\Gamma$-Euler is defined as an integral with respect over orbit space groupoid, $\Gamma$-inertia usual associated groupoid. A key ingredient application o-minimal structures study spaces Our main result that coincide generalize higher-order characteristics Gusein-Zade, Luengo, Melle-Hern\'{a}ndez from case translation groupoid by compact $\Gamma = \mathbb{Z}^\ell$. By realizing space, demonstrate it Morita invariant in category satisfies familiar properties classical characteristic. We give additional formulation for terms finite covering orbispace charts. In abelian extension bundle groups, relate those groups.