Abstract: For the orbifold theorem we have now a complete proof in the vertex case with Leeb and Porti. Using the extension of Thurston hyperbolization therem to Haken orbifolds, we restrict to the case of small orbifolds (those are orbifolds with empty or turnovers boundary, that contain no bad 2-suborbifolds, neither essential 2-suborbifolds). The theorem we are proving then is:
Thm. Small orbifolds are geometric
In the cyclic case that is done in our text with Porti, to appear as a monography of Asterisque. Then, using geometry of hyperbolic cone manifolds, we treate the case where the boundary is not empty.
We reduce the closed dihedral case to the cyclic case, using that a small dihedral orbifold has a finite regular covering of cyclic type.
In the non-dihedral case, the difficult case was the finite fundamental group case. Here we use the idea, given to us by D. Cooper, that if we replace all the branching indices by 2, then we are in the dihedral case, hence the orbifold is spherical. Then we start deacreasing the cone angles and use the study of deformations of spherical cone structures to reach the original orbifold angles. One of the key ingredients is the topological structure of real algebraic curves.