The Moduli Space of Hyperbolic Cone Structures

Abstract

Let $\Sigma$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, \Sigma)$ with all cone angle less than $2\pi /3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corollary, we will give a proof of a special case of Thurston's geometrization theorem for orbifolds.