dg.differential geometry - Diffeomorphism groups of orbifolds

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!).