Branched projective structures with Fuchsian holonomy

Abstract

We prove that if $S$ is a closed compact surface of genus $g\ge 2$, and if $\rho :{\pi }_1\left(S\right)\to PSL\left(2,ℂ\right)$ is a quasi-Fuchsian representation, then the space ${\mathcal{ℳ}}_{k,\rho }$ of branched projective structures on $S$ with total branching order $k$ and holonomy $\rho$ 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 ${\mathcal{ℳ}}_{k,\rho }$ for non-elementary representations $\rho$. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.