Monodromy groups of CP1‐structures on punctured surfaces

For a punctured surface S, we characterize the representations of its fundamental group into PSL 2 ( 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 C 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 PSL 2 ( C ) -representations, following ideas of Thurston and some recent results of Allegretti–Bridgeland.