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Researches on quasiconformal groups and the modular group of the universal Teichmuller space

通用Teichmuller空间的拟共形群和模群研究

基本信息

批准号：
16340036
负责人：
MATSUZAKI Katsuhiko
金额：
$ 10.6万
依托单位：
Okayama University (2006-2007)Ochanomizu University (2004-2005)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

The Teichmuller Tare is a deformation space of the conformal structures of a Riemann surface. The quasiconformal mapping class group is a certain quotient group of the quasiconformal homeomorphisms of the Riemann surface and it acts on the Teichmuller space as the group of biholomorphic automorphisms (modular transformations). When Teichmuller spaces are finite dimensional, they are widely studied with great importance in various fields of mathirnatics. We aim to extend them to infinite dimensional Teichmuller sperms. In this research, we investigated the dynamics of the quasiconformal mapping class on the Teichmuller space. For this purpose, we also considered a certain quotient space of the Teichmuller space, which is called the asymptotic Teichmuller space.We first investigated the recurrent set for the mapping class group and proved that the periodic points are not dense in this set. This result was a foundation of our further studies on the action of elliptic modular transformatio … More n (conformal mapping classes) and the classification of the modular transformations. Our classification was based on the behavior of the orbit and we specified two classes, which have a similar nature of the modular transformations of finite dimensional Teichmuller spaces. One is a class of stationary mapping classes, and the other is a class of modular transformations that have a fixed point on the asymptotic Teichmuller space. We noticed that the action of a stationary mapping class group is stable, but also gave an example of a non-stationary mapping class group that acts discontinuously. As an extreme case, we dealt with a mapping class group that has a common fixed point on the asymptotic Teichmuller space and proved that such a group consists of countably many elements.As another topic, we studied holomorphic self-covering of Riemann surfaces. We gave a necessary condition for a hyperbolic Riemann surface to admit a (non-injective) holomorphic self-cover in terms of the corresponding Fuchisian group. Namely, if the Fuchsian group is of divergence type at the critical exponent of its Poincare series, then the Riemann surface has no self-covers. The proof used uniqueness of the Patterson-Sullivan measure and can be extended to higher dimensional cases. A holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmuller space. We investigated the dynamics of such a self-embedding and examined the distribution of isometric tangent vectors over Teichmuller space. We also extended our observation to quasiregular self-covers of Riemann surfaces. Less

Teichmuller Tare 是黎曼曲面共形结构的变形空间。拟共形映射类群是黎曼曲面拟共形同胚的某个商群，它作为双全纯自同构（模变换）群作用于 Teichmuller 空间。当 Teichmuller 空间是有限维时，它们在数学的各个领域中得到了广泛的研究和重要的意义。我们的目标是将它们扩展到无限维的 Teichmuller 精子。在这项研究中，我们研究了 Teichmuller 空间上拟共形映射类的动力学。为此，我们还考虑了Teichmuller空间的某个商空间，称为渐近Teichmuller空间。我们首先研究了映射类群的循环集，并证明了该集中的周期点不是稠密的。这一结果为我们进一步研究椭圆模变换（共形映射类）的作用和模变换的分类奠定了基础。我们的分类是基于轨道的行为，并且我们指定了两个类，它们具有有限维 Teichmuller 空间的模变换的相似性质。一类是平稳映射类，另一类是在渐近 Teichmuller 空间上有不动点的模变换类。我们注意到平稳映射类组的动作是稳定的，但也给出了非平稳映射类组的不连续动作的示例。作为一个极端情况，我们处理了一个在渐近Teichmuller空间上有公共不动点的映射类群，并证明了这样一个群由可数个元素组成。作为另一个课题，我们研究了黎曼曲面的全纯自覆盖。我们给出了双曲黎曼曲面根据相应的 Fuchisian 群承认（非内射）全纯自覆盖的必要条件。即，如果 Fuchsian 群在其 Poincare 级数的临界指数处是散度型的，则黎曼曲面没有自覆盖。该证明使用了帕特森-沙利文测度的唯一性，并且可以扩展到更高维度的情况。黎曼曲面的全纯自覆盖导致其 Teichmuller 空间的非满射全纯自嵌入。我们研究了这种自嵌入的动力学，并检查了等距切向量在 Teichmuller 空间上的分布。我们还将观察扩展到黎曼曲面的拟正则自覆盖。较少的

项目成果

期刊论文数量（28）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmuller spaces

渐近 Teichmuller 空间上具有公共不动点的拟共形映射类群

DOI：
发表时间：
2007
期刊：
J, d' Analyse Math 102
影响因子：
0
作者：
H. Aikawa;et. al.;K. Matsuzaki
通讯作者：
K. Matsuzaki

The Teichmuller space ofthe ideal boundary

理想边界的Teichmuller空间

DOI：
发表时间：
2006
期刊：
Hiroshima Math. Journal vol.36
影响因子：
0
作者：
Masaharu Nishio;Noriaki Suzuki;Masahiro Yamada;H. Masaoka;H. Masaoka and S. Segawa;M. Taniguchi
通讯作者：
M. Taniguchi

Inclusion relations between the Bers embeddings

Bers 嵌入之间的包含关系