C-gluing construction and slices of quasi-Fuchsian space

Abstract

Given a pants decomposition $\mathcal{P}\mathcal{C}=\{{\gamma }_1,\dots ,{\gamma }_{\xi }\}$ on a hyperbolizable surface $\Sigma$ and a vector $\underline{c}=(c_1,\dots ,c_{\xi })\in {\mathbb{R}}_{+}^{\xi }$, 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 $\underline{c}\to \underline{0}=(0,\dots ,0)$ 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.

27 pages, 10 figures