Postprint version. Published in Infinite Dimensional Lie Groups in Geometry and Representation Theory, January 1, 2002, pages 116-137.
In this paper we outline results on orbifold diffeomorphism groups that were presented at the International Conference on Infinite Dimensional Lie Groups in Geometry and Representation Theory at Howard University, Washington DC on August 17-21, 2000. Specifically, we define the notion of reduced and unreduced orbifold diffeomorphism groups. For the reduced orbifold diffeomorphism group we state and sketch the proof of the following recognition result: Let O1 and O2 be two compact, locally smooth orbifolds. Fix r ≥ 0. Suppose that Φ : Diffr (O1) → Diffr (O2) is a group isomorphism. Then Φ is induced by redred a (topological) homeomorphism h : X→ XThat is, Φ(f)= hfh−1 for O1 O2 . r all f ∈ Diffr (O1). Furthermore, if r>0, his a Cmanifold diffeomorphism red when restricted to the complement of the singular set of each stratum. We then show that if we replace the reduced orbifold diffeomorphism group by the unreduced orbifold diffeomorphism group in the above theorem, we can strengthen the homeomorphism hto an orbifold homeomorphism (orbifold structure preserving). Lastly, we state a structure theorem for the orbifold diffeomorphism group, showing that it is a Banach manifold for 1 ≤ r
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