Geometry and topology of geometric limits I

Details Geometry and topology of geometric limits I Geometry and topology of geometric limits I

2010-02-23

arxiv.org/abs/1002.4266v1

Mathematics

Geometric Topology

Scientific paper

In this paper, we are concerned with hyperbolic 3-manifolds $\hyperbolic^3/G$ such that $G$ are geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we shall show that such a hyperbolic 3-manifold is uniformly bi-Lipschitz homeomorphic to a model manifold which has a structure called brick decomposition and is embedded in $S \times (0,1)$. Conversely, any such 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. Finally, it will be 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, which is analogous to the ending lamination theorem for the case of finitely generated Kleinian groups.

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