Hyperbolic Manifolds and Discrete Groups
Hyperbolic Manifolds and Discrete Groups
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DOI：
10.1007/978-0-8176-4913-5
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发表时间：
2000-11
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影响因子：
--
通讯作者：
M. Kapovich
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文献类型：
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作者：
M. Kapovich
本书的主要目的是证明以下几点。Thurston's Hyperbolization Theorem（Thurston's Hyperbolization Theorem），《大怪物》（The Big Monster）设M是一个紧致的超环面Haken 3-流形，其Euler特征线为零.则M的内部存在一个有限体积的完备双曲度量。这个定理建立了3-流形的几何和拓扑3与Isom（JH [）的离散子群代数之间的强联系。它彻底改变了三维拓扑学和克莱因群理论的面貌。此外，它允许一个证明的东西，超出了标准的3-流形技术，例如，史密斯猜想，剩余有限性的基本群体哈肯流形等，在这本书中，我们提出了一个完整的证明双曲化定理的”一般情况下。“最初，我们计划1包括一个详细的证明，在剩余的情况下，流形纤维超过§以及。然而，由于Otal的书[Ota 96]（其中对待纤维束的情况下）成为可用，只有一个草图的证明在纤维的情况下将在这里。
The main goal of the book is to present a proof of the following. Thurston's Hyperbolization Theorem (" The Big Monster"). Suppose that M is a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the interior of M admits a complete hyperbolic metric of finite volume. This theorem establishes a strong link between the geometry and topology 3 of 3-manifolds and the algebra of discrete subgroups of Isom (JH [). It completely changed the landscape of 3-dimensional topology and theory of Kleinian groups. Further, it allowed one to prove things that were beyond the reach of the standard 3-manifold technique as, for example, Smith's conjecture, residual finiteness of the fundamental groups of Haken manifolds, etc. In this book we present a complete proof of the Hyperbolization Theorem in the" generic case." Initially we planned 1 including a detailed proof in the remaining case of manifolds fibered over § as well. However, since Otal's book [Ota96](which treats the fiber bundle case) became available, only a sketch of the proof in the fibered case will be given here.