Hyperbolicity of links complements in Seifert fibered spaces

Abstract

Let $\bar{\gamma }$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal{O}$ 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 give combinatorial bounds of its volume.

V3: fixed many typos, clarified some arguments and improved expositions