Monodromy groups of CP1?structures on punctured surfaces

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as monodromy meromorphic projective structure on $S$ with poles order at most two and no apparent singularities. This proves analogue theorem Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures closed surfaces, settles long-standing question about characterizing groups for Schwarzian equation spheres. The proof involves geometric interpretation Fock-Goncharov coordinates moduli space framed (\mathbb{C})$-representations, following ideas Thurston some recent results Allegretti-Bridgeland.