Geometry and topology of geometric limits I

TL;DR

This paper classifies hyperbolic 3-manifolds as geometric limits of Kleinian surface groups, constructs bi-Lipschitz models, and establishes end invariants, extending Thurston's questions on hyperbolic geometry.

Contribution

It introduces bi-Lipschitz model manifolds with brick decompositions for hyperbolic 3-manifolds and defines end invariants, advancing understanding of geometric limits.

Findings

Constructed bi-Lipschitz model manifolds with brick decompositions.

Proved that such models are homeomorphic to geometric limits of quasi-Fuchsian groups.

Defined end invariants that determine the isometric type of the manifolds.

Abstract

In this paper, we classify completely hyperbolic 3-manifolds corresponding to 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 construct bi-Lipschitz model manifolds for such hyperbolic 3-manifolds, which have a structure called brick decomposition and are embedded in $S \times (0,1)$. 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, which is analogous to the ending lamination theorem for the case of finitely generated Kleinian groups. These constitute an attempt to give an answer to the 8th question among the 24 questions raised by Thurston in his BAMS paper.