On boundaries of teichmüller spaces and on kleinian groups, III
On boundaries of teichmüller spaces and on kleinian groups, III
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关于 teichmüller 空间的边界和 kleinian 群，III
DOI：
10.1007/bf02392102
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发表时间：
1975
期刊：
影响因子：
3.7
通讯作者：
W. Abikoff
中科院分区：
文献类型：
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作者：
W. Abikoff
标记黎曼曲面 S 的 Teichmfiller 空间，具有签名的有限共形类型 (g, n) 或表示 S 的第一类有限生成的 Fuchsian 群，被 Ahlfors [3] 显示为具有复数结构（另请参见 Rauch [23]）。 Bers [7] 后来证明它可以作为有界域嵌入到 S 上的有界二次微分空间中；该空间的维度为 3g-3 § Teichmfiller 空间的 Bers 嵌入自然适合于有关边界的问题。在两部分几乎联合发表的论文中，Bers [8] 和 Maskit [20] 系统地研究了该边界。特别是，Bers 表明完全退化的克莱因群出现在边界上。完全退化的克莱因群有一个不连续区域，该区域是连通的和单连通的。马斯基特利用他的深层构建技术，详尽地列出了可能会在边界上发生的病理情况，并表明它们确实发生了。我们将研究那些在 w 1 中定义的意义上非病态的组，并且被称为常规组。该类与 Bers 的非简并非准 Fuchsian 群和 Maskit 群给出 S 的完全分解。正则性是一个平面概念，尽管我们将在 w 7 中证明正则性等价于几何有限性。由于我们需要多次参考本系列中的早期论文，因此 Bers [8] 将表示为 BI，Maskit [20] 将表示为 B-II。 Maskit [20] 建设性地研究了非病态边界群的类别，并由 Marden [19] 以及 Earle 和 Marden [12] 在无扭情况下使用其相关的 3 流形结构进行了建设性研究。目前的工作中使用的技术是二维的和非建设性的。它们涉及共形和拟共形映射以及平面拓扑，尽管大部分工作都深深地（如果不是直接地在精神上）借鉴了Maskit的工作。 Fuchsian群空间中进行了相关研究
The Teichmfiller space of a marked Riemann surface S, of finite conformal type (g, n) with signature or of a finitely generated Fuchsian group of the first kind representing S, was shown to have a complex structure by Ahlfors [3](see also Rauch [23]). Bers [7] later showed that it could be embedded as a bounded domain in the space of bounded quadratic differentials on S; this space has dimension 3g-3 § The Bers embedding of the Teichmfiller space lends itself naturally to questions about the boundary. In a two part, almostjoint paper, Bers [8] and Maskit [20] systematically examined that boundary. In particular, Bers showed that totally degenerate Kleinian groups are represented on the boundary. A totally degenerate Kleinian group has a region of discontinuity which is connected and simply connected. Using his deep construction techniques, Maskit gave an exhaustive list of those pathologies which might be expected to occur on the boundary and showed that they do indeed occur. We shall study those groups which are not pathological, in a sense to be defined in w 1, and are called regular. This class coincides with Bers' non-degenerate non-quasi-Fuchsian groups and Maskit's groups giving complete factorizations of S. Regularity is a planar concept although we will show in w 7 that regularity is equivalent to geometric finiteness.Since we will need to refer on many occasions to the earlier papers in this series, Bers [8] will be denoted BI and Maskit [20] will be denoted B-II. The class of non-pathological boundary groups has been studied constructively by Maskit [20] and using the structure of their associated 3-manifolds, in the torsionfree case, by Marden [19] and Earle and Marden [12]. The techniques used in the present work are twodimensional and non-constructive. They involve conformal and quasiconformal mappings and plane topology, although much of the work draws deeply, if not directly then in spirit, on the work of Maskit. Related studies in the space of Fuchsian groups have been conducted
克莱因群的自守函数。
DOI：
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发表时间：
2008
期刊：
Geometry \& Topology Monographs 13
影响因子：
--
作者：
M.Yoshida;T.Sasaki;H. Nakada;林仲夫;M.Yoshida
通讯作者：
M.Yoshida