Hyperbolicity of link complements in Seifert-fibered spaces

Abstract

Let $\bar{\gamma }$ be a link in a Seifert-fibered space $M$ over a hyperbolic $2$–orbifold $\mathcal{𝒪}$ that projects injectively to a filling multicurve of closed geodesics $\gamma$ in $\mathcal{𝒪}$. We prove that the complement $M_{\bar{\gamma }}$ of $\bar{\gamma }$ in $M$ admits a hyperbolic structure of finite volume, and we give combinatorial bounds of its volume.