# Meromorphic projective structures, grafting and the monodromy map

## TL;DR

This paper extends Thurston's grafting theorem to meromorphic projective structures with poles of order three or more, linking them to crowned hyperbolic surfaces and analyzing their monodromy maps.

## Contribution

It proves a grafting theorem for meromorphic projective structures with poles of order three or more, and shows the monodromy map is a local homeomorphism.

## Key findings

- Grafting theorem extended to meromorphic structures
- Monodromy map is a local homeomorphism
- Provides a grafting description for polynomial Schwarzian derivatives

## Abstract

A meromorphic projective structure on a punctured Riemann surface $X\setminus P$ is determined, after fixing a standard projective structure on $X$, by a meromorphic quadratic differential with poles of order three or more at each puncture in $P$. In this article we prove the analogue of Thurston's grafting theorem for such meromorphic projective structures, that involves grafting crowned hyperbolic surfaces. This also provides a grafting description for projective structures on $\mathbb{C}$ that have polynomial Schwarzian derivatives. As an application of our main result, we prove the analogue of a result of Hejhal, namely, we show that the monodromy map to the decorated character variety (in the sense of Fock-Goncharov) is a local homeomorphism.