# Monodromy representations of meromorphic projective structures

## TL;DR

This paper characterizes the monodromy representations of meromorphic projective structures with high-order poles, extending classical results and utilizing advanced moduli space coordinates.

## Contribution

It proves the monodromy map image for structures with poles of order >2, answering longstanding questions and generalizing Gallo-Kapovich-Marden's theorem.

## Key findings

- Determined the monodromy map image for meromorphic structures with high-order poles
- Extended classical monodromy theorems to meromorphic cases
- Utilized Fock-Goncharov coordinates for moduli space analysis

## Abstract

We determine the image of the monodromy map for meromorphic projective structures with poles of orders greater than two. This proves the analogue of a theorem of Gallo-Kapovich-Marden, and answers a question of Allegretti and Bridgeland. Our proof uses coordinates on the moduli space of framed representations arising from the work of Fock and Goncharov.