Hyperbolic 3-manifolds with totally geodesic boundary (Q803587)

For an integer \(g\geq 2\) let \(M_ g\) be the Riemann moduli space of isometry classes of closed hyperbolic surfaces of genus g. A claim by Thurston is that the subset S of \(M_ g\) consisting of those surfaces which bound compact hyperbolic 3-manifolds (with totally geodesic boundaries) is a countably infinite dense subset of \(M_ g\). The author constructs for each fixed \(g\geq 2\) infinitely many non-isometric compact oriented hyperbolic 3-manifolds with totally geodesic boundary of genus g. He conjectures that the boundary surfaces in these examples have different moduli.