A Schottky decomposition theorem for complex projective structures

Abstract

Let $S$ be a closed orientable surface of genus at least two, and let $C$ be an arbitrary (complex) projective structure on $S$. We show that there is a decomposition of $S$ into pairs of pants and cylinders such that the restriction of $C$ to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on $\left(S,C\right)$, along which a grafting can be done.