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

Teichmuller Tare是黎曼曲面的共形结构的变形空间。拟共形映射类群是黎曼曲面的拟共形同胚的某个商群，它作为双全纯自同构（模变换）的群作用在Teichmuller空间上。当Teichmuller空间是有限维时，它们在数学的各个领域都有着重要的研究。我们的目标是将它们扩展到无限维的Teichmuller精子。本文研究了Teichmuller空间上拟共形映射类的动力学性质。为此，我们还考虑了Teichmuller空间的一个商空间，称之为渐近Teichmuller空间，并首先研究了映射类群的回归集，证明了在这个回归集中周期点是不稠密的。这一结果为进一步研究椭圆模变换的作用奠定了基础. ...更多信息 n（保角映射类）和模变换的分类。我们的分类是基于轨道的行为，我们指定了两个类，它们具有类似的性质的有限维Teichmuller空间的模变换。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，一个平稳的映射类组的作用是稳定的，但也给出了一个例子，一个非平稳的映射类组的作用是不连续的。作为一种极端情况，我们讨论了在渐近Teichmuller空间上有公共不动点的映射类群，并证明了这样的群由可数个元素组成。给出了双曲Riemann曲面在相应的Fuchisian群下存在（非内射）全纯自覆盖的一个必要条件.也就是说，如果Fuchsian群在其Poincare级数的临界指数处是发散型的，则黎曼曲面没有自覆盖。证明使用Patterson-Sullivan测度的唯一性，并且可以扩展到高维情况。黎曼曲面的全纯自覆盖诱导其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，并研究了Teichmuller空间上等距切向量的分布。我们还将我们的观察推广到黎曼曲面的拟正则自覆盖。少

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