Tame and relatively elliptic CP1-structures on the thrice-punctured sphere
Samuel A. Ballas, Philip L. Bowers, Alex Casella, Lorenzo Ruffoni
Suppose a relatively elliptic representation $\rho$ of the fundamental group of the thrice-punctured sphere S is given. We prove that all projective structures on S with holonomy $\rho$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $\rho$ . In the process, we show that (on a general surface $\Sigma$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Möbius completion, and in terms of certain meromorphic quadratic differentials.