Tame and relatively elliptic ${\mathbb{CP}}^1$-structures on the thrice-punctured sphere

Abstract

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.

58 pages, 19 figures, comments welcome. v2: added remarks 3.3.8 and 3.3.9; updated references; fixed minor typos and imprecisions. Final version to appear on Algebraic & Geometric Topology

References in corpus (1)

• Translation surfaces and periods of meromorphic differentials

Cited by in corpus (2)

• Monodromy of Schwarzian equations with regular singularities
• Infinite circle packings on surfaces with conical singularities