Gluing of orbifolds Suppose that $P$ and $Q$ are $n$-dimensional orbifolds, with boundaries. Suppose also that there is an isomorphism $f \colon \partial P \rightarrow \partial Q$ (as orbifolds). Is there a way to glue $P$ and $Q$ using $f$ to form a new closed orbifold? This is an analogy with manifolds with boundary.
 A: Ok so turning my comments into an answer (which I am honestly still only 95% sure about but hopefully some will point out if there is an error).
Lets assume that the two charts about our boundary point are of the form $[0,\infty) \times \mathbb{R}^{n-1}$ with finite groups $G_{i}$ acting, each preserving $\{0\} \times \mathbb{R}^{n-1}$.
We may assume that $G_{i}$ acts effectively and linearly on each chart, respectively. This implies that the induced action on $\{0\} \times \mathbb{R}^{n-1}$ is effective (since the chart has boundary it is not possible to do a reflection in the boundary hyperplane).
By assumption the two orbifolds $(\{0\} \times \mathbb{R}^{n-1},G_{i})$ are isomorphic. That is we have $G_{1} = G_{2}$ (because both of the actions are effective) and that there is a homeomorphism of $(\{0\} \times \mathbb{R}^{n-1})$ respecting the $G_i$ actions. This allows us to glue the two half spaces equivariantly creating an orbifold chart $(\mathbb{R}^{n}, G_{1} = G_{2})$, as required.
