has no relation to 4 is the following : 5. (the virtually fibered conjecture) Any hyperbolic 3–manifold of finite volume has a finite cover which is a surface bundle over the circle. The space of all hyperbolic 3–manifolds Geometric convergence A sequence of Kleinian groups is said to be geometrically convergent if it converges in the Chabauty topology. For the manifolds obtained as quotients this amounts to them being convergent in the pointed Gromov-Hausdorff metric. Jørgensen–Thurston theory The hyperbolic volume can be used to order the space of all hyperbolic manifold. The set of manifolds corresponding to a given volume is at most finite, and the set of volumes is well-ordered and of order type . More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with cusps is a limit of a sequence of manifolds with cusps for any , so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem also proves that any convergent sequence must be obtained by Dehn surgeries on the limit manifold. Quasi-Fuchsian groups Sequences of quasi-fuchsian surface groups of given genus can converge to a doubly degenerate surface group, as in the double limit theorem. Notes References 3-manifolds Hyperbolic geometry Kleinian groups Manifolds Riemannian manifolds