Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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S. A. Ballas - P. L. Bowers - A. Casella - L. Ruffoni

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

created by ruffoni on 01 Jun 2022
modified on 27 Jul 2023


Accepted Paper

Inserted: 1 jun 2022
Last Updated: 27 jul 2023

Journal: Algebraic and Geometric Topology
Year: 2021

ArXiv: 2107.06370 PDF


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 M\"obius completion, and in terms of certain meromorphic quadratic differentials.

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