Determination of the topological structure of an orbifold by its group of orbifold diffeomorphisms

We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let Diff _Orb^r(O) denote the C^r orbifold diffeomorphisms of an orbifold O. Suppose that Φ: Diff_Orb^r(O ₁) → Diff_Orb^r(O₂) is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds O₁ and O₂. We show that Φ is induced by a homeomorphism h: X_O1 → X_O2, where X_O denotes the underlying topological space of O. That is, Φ(f) = hfh ^-1 for all f ∈ Diff_Orb^r(O₁). Furthermore, if r > 0, then h is a C^r manifold diffeomorphism when restricted to the complement of the singular set of each stratum.