Branched projective structures with quasi-Fuchsian holonomy
  We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for the space M(k,R). It is shown to be a smooth complex manifold modeled on Hurwitz spaces. 
Gabriel Calsamiglia, Bertrand Deroin, Stefano Francaviglia
math.GT math.CV