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Determination of the topological structure of an orbifold by its group of orbifold diffeomorphisms. (English) Zbl 1029.58004
Summary: We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let $$\text{Diff}^r_{\text{Orb}}({\mathcal O})$$ denote the $$C^r$$ orbifold diffeomorphisms of an orbifold $${\mathcal O}$$. Suppose that $$\Phi: \text{Diff}^r_{\text{Orb}}({\mathcal O}_1)\to \text{Diff}^r_{\text{Orb}}({\mathcal O}_2)$$ is a group isomorphism between the orbifold diffeomorphism groups of two orbifolds $${\mathcal O}_1$$ and $${\mathcal O}_2$$. We show that $$\Phi$$ is induced by a homeomorphism $$h: X_{{\mathcal O}_1}\to X_{{\mathcal O}_2}$$, where $$X_{{\mathcal O}}$$ denotes the underlying topological space of $${\mathcal O}$$. That is, $$\Phi(f)= hfh^{-1}$$ for all $$f\in \text{Diff}^r_{\text{Orb}}({\mathcal O}_1)$$. Furthermore, if $$r> 0$$, then $$h$$ is a $$C^r$$ manifold diffeomorphism when restricted to the complement of the singular set of each stratum.

##### MSC:
 58D19 Group actions and symmetry properties 57S25 Groups acting on specific manifolds 53C20 Global Riemannian geometry, including pinching
##### Keywords:
orbifold diffeomorphism group
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