Geometry and Topology of Geometric Limits I

In this chapter, we classify completely, up to isometry, hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to π 1(S) for a finite-type hyperbolic surface S. In the first of the three main theorems which constitute the basic results of this chapter, we construct bi-Lipschitz model manifolds for such hyperbolic 3-manifolds, which have a structure called brick decomposition and are embedded topologically in S × (0, 1). In the second theorem, we show that conversely, any such model manifold admitting a brick decomposition with reasonable conditions is bi-Lipschitz homeomorphic to a hyperbolic manifold corresponding to some geometric limit of quasi-Fuchsian groups. In the third theorem, it is shown that we can define end invariants for hyperbolic 3-manifolds appearing as geometric limits of Kleinian surface groups, and that the homeomorphism type and the end invariants determine the isometric type of a manifold. This is analogous to the ending lamination theorem for the case of finitely generated Kleinian groups. These results constitute an attempt to give an answer to the 8th question among the famous 24 questions raised by Thurston.