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 ?


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!).

  • $\begingroup$ Thanks for this quick and illuminating answer. Do you know if some of those results extend to non-hyperbolic orbifolds ? $\endgroup$ – Martin Pinsonnault Jul 7 '10 at 19:51
  • $\begingroup$ They fail even in the classical case for non-hyperbolic surfaces (for instance, for the sphere or the torus). $\endgroup$ – Andy Putman Jul 7 '10 at 19:54
  • $\begingroup$ 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 ? $\endgroup$ – Martin Pinsonnault Jul 7 '10 at 19:58
  • $\begingroup$ Sorry, I'm not aware of any such results. I'd be interesting in hearing about any ones that you find! $\endgroup$ – Andy Putman Jul 7 '10 at 20:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.