On limits of quasi-conformal deformations of Kleinian groups
On limits of quasi-conformal deformations of Kleinian groups
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关于克莱因群拟共形变形的极限
DOI：
10.1007/bf01160674
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发表时间：
1989
影响因子：
0.8
通讯作者：
Ken'ichi Ohshika
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作者：
Ken'ichi Ohshika
令 F 为有限生成的非初等克莱因群（第二类）。令 Ar 表示其极限集，并令 f2 r 表示其不连续区域。假设 f2 r 的所有分量都是单连接的。 Bers [4] 和 Sullivan [21] 等人的工作表明，F 的拟共形变形空间与 (2r/F 的 Teichm/F 空间，用 J-(f2r/F) 表示) 之间存在同胚。一个简单的例子是当 F 是属于 g 的第一类无挠 Fuchsian 群时的情况。则 ~ 2 r 是两个单连通的并集 组件。因此，与 F 同构的拟 Fuchsian 群的空间同构于 Jg x Wg。一个自然的问题是，当一系列克莱因群通过上述同胚在 J (f2r/F) 中趋向无穷大时，会发生什么。第一个例子是 Bers ([-3]) 研究的准 Fuchsian 群。在那里他证明了一系列 对应于 {m0 x m'} c~ x Jgg 的准 Fuchsian 群，其中 mo 是固定的，m'i 在 ~ 中无穷大，有一个代数收敛于克莱因群的子序列，该克莱因群仅具有 f2 r 的一个不变分量。第二个例子是瑟斯顿的“双重极限定理”（[-24]）。它指出，对于两个测量的叠片 2， 2' 满足一定条件，对应于被视为 Teichm/iller 空间 Thurston 边界元素的收敛于 (2, 2') 的序列的拟 Fuchsian 群序列，具有代数收敛于 Kleinian 群（第一类）的子序列。另一方面，瑟斯顿在[23]中证明了圆柱双曲的变形空间 代数拓扑的 3 流形是紧的。因此，在这种情况下，每个序列都有一个代数收敛于克莱因群的子序列。
Let F be a finitely generated non-elementary Kleinian group (of the second kind). Let Ar denote its limit set and let f2 r denote its region of discontinuity. Assume that all components of f2 r are simply connected. The work of Bers [4] and Sullivan [21], among others, has shown that there is a homeomorphism between the space of quasi-conformal deformations of F and the Teichm/iller space of (2r/F denoted by J-(f2r/F). A simple example is the case when F is a torsion free Fuchsian group of first kind with genus g. Then~ 2 r is a union of two simply connected components. Therefore the space of quasi-Fuchsian groups which are isomorphic to F is homeomorphic to Jg x Wg. It is a natural question to ask what happens when a sequence of Kleinian groups goes to infinity in J (f2r/F) through the homeomorphism above. The first example is that of quasi-Fuchsian groups studied by Bers ([-3]). There he has shown that a sequence of quasi-Fuchsian groups corresponding to {m0 x m'} c~ x Jgg, where mo is fixed and m'i goes to infinity in~, has a subsequence converging algebraically to a Kleinian group which has only one invariant component of f2 r. The second example is" the double limit theorem" of Thurston ([-24]). It states that, for two measured laminations 2, 2'satisfying a certain condition, a sequence of quasi-Fuchsian groups corresponding to a sequence converging to (2, 2') regarded as elements of the Thurston boundary of Teichm/iller space has a subsequence converging algebraically to a Kleinian group (of the first kind). On the other hand Thurston proved in [23] that deformation spaces of acylindrical hyperbolic 3-manifolds with the algebraic topology are compact. Hence in this case every sequence has a subsequence converging algebraically to a Kleinian group.
3 流形群对 R 树的小作用和双曲结构的退化
DOI：
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发表时间：
2006
期刊：
影响因子：
--
作者：
K.Ohshika;H.Miyachi;Haruko NISHI(高山晴子);Haruko Nishi;Ken'ichi Ohshika;Ken'ichi Ohshika;Ken'ichi;Ken'ichi Ohshika;大鹿 健一;大鹿 健一;Ken'ichi Ohshika;Ken'ichi Ohshika;Ken'ichi Ohshika
通讯作者：
Ken'ichi Ohshika