TEICHMULLER SPACES AND GEOMETRY OF MOBIUS TRANSFORMATIONS

我在这学期的研究主要包括以下三个分支：1。角度参数与莫比乌斯变换几何关系的思考。用角度参数表示Teichmuller模群（映射类群）。黎曼曲面上简单分环的解析表征。为了得到Teichmuller空间的全局真实解析和简单表示，我引入了新的角度参数。证明了Theichmuller空间仅由角度参数描述，分析典型的Teichmuller空间的这种角度参数空间很容易。考虑到发生器的轴和这些Fuchsian群的产物，例如曾经有孔的环面Fuchsian群，我发现了这些轴排列的高度对称性。利用莫比乌斯变换的1 / 2次幂和双曲几何，研究了莫比乌斯变换、轨迹和角度参数之间的关系。从这些观测结果中，得到了角度参数之间的关系和信息。根据这些角度参数的信息，我尝试只用角度参数来表示Teichmuller模群。我考虑了以下几点：(1)代表这些组的角度参数和长度参数之间的关系。(2)在易于计算的情况下，仅用角度参数对该类群进行具体描述。(3)选择简单表示一般情况下Theichmuller模群的角度参数（归纳）。我特别研究了一次孔环面和紧致2属黎曼曲面的Theichmuller模群的表示。进一步，利用表示S的Fuchsian群G对特殊线性群SL （2， C）的升力，给出了黎曼曲面S上简单环路L可分的充分必要条件。例如，如果S是p（>1）属的紧致黎曼曲面，则得到：G的升力数为2的p次幂。设g是g对应于L的一个元素，那么L要除当且仅当对于g的任何升力，g对应的矩阵总是有负迹。少

My research during this term consists mainly of the following three branches :1. Consideration of the relation between angle parameters and the geometry of Mobius transformations.2. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.3. Characterization of simple dividing loops on Riemann surfaces analytically.In order to obtain global real analytic and simple representations of the Teichmuller spaces, I introduced new angle parameters. I showed that the Theichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces.Considering the axes of the generators and these products of Fuchsian groups, for example the once-holed torus Fuchsian groups, I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters, using the one-half powers of Mobius transformations an … More d the hyperbolic geometry. From these observations, the much relation and information of angle parameters were obtained. From such information of angle parameters, I tried to represent the Teichmuller modular groups by only angle parameters. I considered the following :(1) Relation between angle parameters and length parameters representing these groups.(2) Concrete description of such groups by only angle parameters in the cases that it is easy to calculate.(3) Choice of angle parameters (inductively) that simply represent the Theichmuller modular groups of the general cases.I especially studied the representation of the Theichmuller modular groups of a once-holed torus and a compact Riemann surface of genus 2.Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, using the lifts of a Fuchsian group G representing S to the special linear group SL (2, C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L.Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace. Less