Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles

Abstract

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy $\rho:\pi_1(S)\to$ PSL$_2\mathbb{C}$ can be joined to the corresponding uniformizing structure $\sigma_\rho$ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.