GROUP-COMPLETIONS AND LIMIT-SETS OF KLEINIAN-GROUPS
GROUP-COMPLETIONS AND LIMIT-SETS OF KLEINIAN-GROUPS
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DOI：
10.1007/bf01418926
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发表时间：
1980-01-01
影响因子：
3.1
通讯作者：
FLOYD, WJ
中科院分区：
文献类型：
--
作者：
FLOYD, WJ
在拓扑学和几何学中，有限生成群和有限生成群的作用经常出现。虽然最重要的例子可能是紧致流形的基本群，但涉及有限生成群的问题也出现在变换群、动力系统和Klein群中。一个非常引人注目的例子是Mostow定理[17]，它说，对于大于或等于3的维度，闭双曲流形由它们的基本群决定到等距。Jakob Nielsen在一系列文章([18-21])中使用双曲2-空间的Poincar6圆盘模型H2作为INT(D2)来研究曲面及其微分同态。给定亏格g>；2的闭曲面M2和微分同胚f：M--+M，他将f提升为同胚f‘：H2~He，并证明了f扩张到圆S 1=P，D2的同胚.此外，该圆上的映射不依赖于特殊的微分同胚Ji，而仅取决于它的同伦型.Nielsen利用H~(M)和Aut(HI(M))对D2的扩张以及H~(M)和Aut(HI(M))对COL‘的响应作用，系统地研究了曲面微分同态的拓扑性质。Mostow定理的证明还利用Hi(M“)在S”-给定两个具有11>；3的闭双曲n-流形M“和N”上的作用以及同构Q>；：Hi(M)-，Ill(N)，存在同伦等价f：M-+N诱导4>；(因为M和N是K(H，1)‘S)。F可以举起到万能盖子来了！“H“~H”，f扩张为同胚f‘：S“-i-+S”-1。M_9Ostow S证明的实质是证明f‘是保形的，这是通过证明J“是拟保形的，然后利用H~(M)对S作用的遍历性”-1来证明保形的。Marguis意识到，至少Mostow认识到用于构造f‘的同伦等价f：M~N不是
Finitely generated groups and actions of finitely generated groups often come up in studying topology and geometry. While the most important example may be as fundamental groups of compact manifolds, questions involving finitely generated groups also arise in transformation groups, dynamical systems, and Kleinian groups. A very striking example is Mostow's theorem [17], which says that, for dimensions greater than or equal to three, closed hyperbolic manifolds are determined up to isometry by their fundamental groups. Jakob Nielsen, in a series of papers ([18-21]), used the Poincar6 disk model of hyperbolic 2-space, H 2, as int (D 2) to study surfaces and their diffeomorphisms. Given a closed surface M 2 of genus g> 2 and a diffeomorphism f: M--+ M, he lifted f to a homeomorphism f': H2~ H e and showed that f extends to a homeomorphism of the circle S 1= P, D 2. Furthermore, the map on the circle does not depend on the particular diffeomorphism J; but only on its homotopy type. Nielsen made use of the extension off'to D 2 and of the COl'-responding actions of H~(M) and Aut (HI (M)) on S 1 to systematically study topological properties of diffeomorphisms of surfaces. The proof of Mostow's theorem also uses the action of Hi (M") on S"-Given two closed hyperbolic n-manifolds M" and N" with 11> 3 and an isomorphism q>: HI (M)--, Ill (N), there is a homotopy equivalence f: M--+ N inducing 4>(since M and N are K (H, 1)'s). f can be lifted to the universal covers to!" H"~ H", and f extends to a homeomorphism f': S"-I--+ S"-1. The essence of M 9 ostow s proof is to show that f'is conformal; this is done by first showing lhat J" is quasi-conformal and then using ergodicity of the action of H~(M) on S"-1 to show conformality. It was realized by Margulis and at least implicity by Mostow that the homotopy equivalence f: M~ N used to construct f'was not