Totally geodesic boundaries are dense in the moduli space (Q1389882)

Let \(F=\Sigma_1\sqcup\dots\sqcup\Sigma_t\) be a closed, oriented surface such that the genus of each component of \(F\) is greater than one. Let \({\mathcal M}(F)={\mathcal M}(\Sigma_1)\times\dots\times{\mathcal M}(\Sigma_t)\) be the moduli space of \(F\). Let \({\mathcal R}(F)\) be the subset of \({\mathcal M}(F)\) consisting of those hyperbolic structures \(s\in{\mathcal M}(F)\) such that there exists a compact, connected oriented hyperbolic 3-manifold \(M\) with totally geodesic boundary \(\partial M\) together with an orientation-preserving isometry \(\phi:\partial M\to F(s)\). Then \({\mathcal R}(F)\) is dense in \({\mathcal M}(F)\). \textit{C. McMullen}'s work [Invent. Math. 99, No. 2, 425-454 (1990; Zbl 0695.57012)] plays an important role in the argument.