(joint work with Ryan Carroll, Whitney DuVal, Carla Farsi, John Schulte, and Bradford Taylor)

If $Q$ is an orbifold presented by an orbifold groupoid $\mathcal{G}$ and $\Gamma$ is a finitely generated group, then $\Gamma$ induces a decomposition of $Q$ into the $\Gamma$-sectors of $Q$, a finite disjoint union of orbifolds including $Q$ as well as lower-dimensional orbifolds related to the singular strata of $Q$. Specifically, the $\Gamma$-sector decomposition is given by the space of groupoid homomorphisms from $\Gamma$ into $\mathcal{G}$, which admits a natural $\mathcal{G}$-action. This construction, which generalizes the inertia orbifold and orbifold of multi-sectors, was introduced by Tamanoi for global quotient orbifolds, i.e. orbifolds given by the quotient of a manifold by a finite group. Applying an orbifold invariant $\varphi$ to the space of $\Gamma$-sectors of $Q$ yields the $\Gamma$-extension $\varphi_\Gamma$ of $\varphi$, a new invariant for orbifolds associated to each $\Gamma$.

We will discuss the $\Gamma$-extensions of the Euler-Satake characteristic. In particular, we will consider these invariants for $2$- and $3$-dimensional orbifolds as well as for wreath symmetric products of orbifolds, the orbifold analogue of the symmetric product of a manifold.