# Diffeomorphism groups of orbifolds

A lot is known about geometric and topological properties of diffeomorphism groups of surfaces (here, I am mainly thinking about the work of Smale and Eells-Elworthy). Is there anything known for orbisurfaces ? My first guess would be that many of these groups must have contractible components since singular points impose extra conditions somewhat similar to fixed points. Is there a good reference on this topic ?

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The result you want can be found in the following paper:

MR0955816 (89h:30028) Earle, Clifford J.(1-CRNL); McMullen, Curt(1-MSRI) Quasiconformal isotopies. Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 143--154, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.

What they prove is actually pretty remarkable. Namely, let $S$ be a hyperbolic surface. Then there is a family $\phi_t$ of self-maps of $\text{Diff}^{0}(S)$ such that $\phi_0$ is the identity, such that $\phi_1$ is the constant map taking each diffeomorphism to the identity diffeomorphism, and such if $f \in \text{Diff}^0(S)$ commutes with a finite order diffeomorphism $g$ of $S$, then $\phi_t(f)$ also commutes with $g$ for all $t$. In other words, you can contract $\text{Diff}^0(S)$ in way that doesn't break any symmetries.

Now assume that $\Sigma = S / \Gamma$ is a good hyperbolic orbifold, where $\Gamma$ is a finite group of diffeomorphisms of $S$. The identity component of the orbifold diffeomorphism group of $\Sigma$ is then homeomorphic to $$\text{Diff}^{0}(S,\Gamma) := \langle f \in \text{Diff}^{0}(S)\ |\ gfg^{-1}=f\ \text{for all}\ g \in \Gamma \rangle \subset \text{Diff}^{0}(S)$$ The null-homotopy $\phi_t$ preserves $\text{Diff}^{0}(S,\Gamma)$, so it is contractible.

(EDIT : I made a slight fix to the definition of the orbifold diffeomorphism above. It doesn't change the argument. Thanks to Tom Church for pointing it out to me!).

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Thanks for this quick and illuminating answer. Do you know if some of those results extend to non-hyperbolic orbifolds ? –  Martin Pinsonnault Jul 7 '10 at 19:51
They fail even in the classical case for non-hyperbolic surfaces (for instance, for the sphere or the torus). –  Andy Putman Jul 7 '10 at 19:54
Well, my question was badly formulated. Here's a second try: Do you know any results on the homotopy type of diffeomorphism groups of non-hyperbolic orbisurfaces (à la Smale), regardless of the techniques used ? –  Martin Pinsonnault Jul 7 '10 at 19:58
Sorry, I'm not aware of any such results. I'd be interesting in hearing about any ones that you find! –  Andy Putman Jul 7 '10 at 20:11