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

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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