Geometry and Topology of Geometric Limits I
Geometry and Topology of Geometric Limits I
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几何极限的几何与拓扑 I
DOI：
10.1007/978-3-030-55928-1_9
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Soma Teruhiko
中科院分区：
文献类型：
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作者：
Ohshika Ken’ichi;Soma Teruhiko
在这一章中，我们对有限类型双曲曲面上与π1(S)同构的Kleian曲面群的几何极限对应的双曲3-流形进行了完全的分类，直到等距。在构成本章基本结果的三个主要定理中的第一个中，我们构造了这类双曲三维流形的双Lipschitz型流形，它具有一种称为砖块分解的结构，并且在拓扑上嵌入Ins× (0，1)。在第二个定理中，我们证明了反之，任何这样的允许合理条件下砖块分解的模型流形都是双Lipschitz同胚的，对应于拟Fuchsian群的某些几何极限的双曲流形。在第三个定理中，我们证明了对于表现为Kleian曲面群的几何极限的双曲三维流形，我们可以定义末端不变量，并且同胚型和末端不变量决定了流形的等距类型。这类似于有限生成Klein群的结束分层定理。这些结果试图回答瑟斯顿提出的著名24个问题中的第8个问题。
In this chapter, we classify completely, up to isometry, hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic toπ1(S) for a finite-type hyperbolic surfaceS. 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 inS× (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.