# Wild globally hyperbolic maximal anti-de Sitter structures

## TL;DR

This paper introduces wild globally hyperbolic anti-de Sitter structures on punctured surfaces, providing two new parameterizations of their deformation space involving Teichmüller spaces and meromorphic quadratic differentials.

## Contribution

It presents the first definitions and parameterizations of wild globally hyperbolic anti-de Sitter structures on punctured surfaces, linking them to Teichmüller theory and meromorphic differentials.

## Key findings

- Two parameterizations of the deformation space are established.
- The deformation space is described as a quotient of Teichmüller spaces.
- The structure involves meromorphic quadratic differentials with poles of order at least 3.

## Abstract

Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichm\"uller space of crowned hyperbolic surfaces and as the bundle over the Teichm\"uller space of $\Sigma$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.