Branched projective structures with Fuchsian holonomy

We prove that if S is a closed compact surface of genus g≥2, and if ρ:π1(S)→PSL(2,ℂ) is a quasi-Fuchsian representation, then the space Mk,ρ of branched projective structures on S with total branching order k and holonomy ρ is connected, for 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 Mk,ρ for non-elementary representations ρ. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.