# C-gluing construction and slices of quasi-Fuchsian space

## TL;DR

This paper introduces a new plumbing construction for hyperbolizable surfaces that produces quasi-Fuchsian structures with prescribed lengths, connecting to Kra's construction and exploring slices of quasi-Fuchsian space.

## Contribution

It presents a novel plumbing method to generate quasi-Fuchsian structures with specified boundary lengths, linking to existing constructions and analyzing special slices of the space.

## Key findings

- Construction limits to Kra's plumbing as lengths approach zero
- Holonomy representations lie in the linear slice of quasi-Fuchsian space for genus one surfaces
- Conjectures proposed for the structure of these slices based on visualizations

## Abstract

Given a pants decomposition $\mathcal{PC} = \{\gamma_1, \ldots, \gamma_{\xi}\}$ on a hyperbolizable surface $\Sigma$ and a vector $\underline{c} = (c_1, \ldots, c_{\xi}) \in \mathbb{R}_+^\xi$, we describe a plumbing construction which endows $\Sigma$ with a complex projective structure for which the associated holonomy representation $\rho$ is quasi-Fuchsian and for which $\ell_\rho(\gamma_i) = c_i$. When $\underline{c} \to \underline{0} = (0, \ldots, 0)$ this construction limits to Kra's plumbing construction. In addition, when $\Sigma = \Sigma_{1,1}$, the holonomy representations of these structures belong to the `linear slice' of quasi-Fuchsian space $\mathrm{QF}(\Sigma)$ defined by Komori and Parkonnen. We discuss some conjectures for these slices suggested by the pictures we created in joint work with Yamashita.