Free and free abelian Euler–Satake characteristics of nonorientable 2-orbifolds

We compute the Γ-sectors and Γ-Euler–Satake characteristic of a closed, effective 2-dimensional orbifold Q where Γ is a free or free abelian group. Using this information, we determine a family of orbifolds such that the complete collection of Γ-Euler–Satake characteristics associated to free and free abelian groups determines the number and type of singular points of Q as well as the Euler characteristic of the underlying space. Additionally, we show that any collection of these groups whose Euler–Satake characteristics determine this information contains both free and free abelian groups of arbitrarily large rank. It follows that the collection of Euler–Satake characteristics associated to free and free abelian groups constitute a finer orbifold invariant than the collection of Euler–Satake characteristics associated to free groups or free abelian groups alone.