Monodromy groups of $\mathbb{C}\mathrm{P}^1$-structures on punctured surfaces

Abstract

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