Analysis and Geometry of the Teichmuller Spaces

我在这学期的研究主要包括以下三个分支：1。黎曼曲面上简单分环的解析表征。用角度参数解析表示Teichmuller空间的全局实数。用角度参数表示Teichmuller模群（映射类群）。我描述了莫比乌斯变换的几何特征通过使用这些变换的1 / 2次方和这些轨迹。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上一个简单环路L可除的充分必要条件。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。为了得到Teichmuller空间的全局实解析和简单表示，在标记的Riemann曲面上引入了与测地线交角对应的新的角度参数。我证明了Teichmuller空间仅由角度参数描述，并且很容易分析（1,1）、（2,0）和（3,0）类型的典型Teichmuller空间的这种角度参数空间。角度参数对应于发电机轴线与标记的Fuchsian组的这些产品之间的交角。我发现了这些轴排列的高度对称性。我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。接下来，我考虑了仅用角度参数表示的Teichmuller模群。我特别研究了以下几点：解释Teichmuller模群是一些特殊的双曲多边形对其他多边形的轴界作用。代表这些组的角度参数和长度参数之间的关系。少

My research during this term consists mainly of the following three branches :1. Characterization of simple dividing loops on Riemann surfaces analytically.2. Representation of the Teichmuller spaces global real analytically by angle parameters.3. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.I Characterized the geometry of Mobius transformations by using the one-half powers of these transformations and these traces. Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, by 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.I in … More troduced new angle parameters corresponding to the intersection angles between geodesics on the marked Riemann surface, in order to obtain global real analytic and simple representations of the Teichmuller spaces. I showed that the Teichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces of types (1,1), (2,0) and (3,0). Angle parameters correspond to the intersection angles between the axes of the generators and these products of the marked Fuchsian group. 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. From these observations, the much relation and information of angle parameters were obtained.Next, I considered the representations of the Teichmuller modular groups by only angle parameters. I especially studied the following :I. Interpretation of the Teichmuller modular groups as the actions of some special hyperbolic polygons bounded by the axes to others.II. Relation between angle parameters and length parameters representing these groups. Less