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Analysis and Geometry of the Teichmuller Spaces

Teichmuller空间的分析和几何

基本信息

批准号：
13640164
负责人：
OKUMURA Yoshihide
金额：
$ 2.18万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640164/
关键词：
Teichmuller space Teichmuller modular group discrete group Riemann surface Mobius transformation simple dividing loop hyperbolic manifold real analytic manifold 解散群

项目摘要

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

本学期的研究主要包括以下三个方面：1.黎曼曲面上简单分割环的解析刻画。用角度参数解析地表示TeichMuller空间的整体实数。用角度参数表示Teichmuller模群(映射类群)。我用这些变换的一半幂和这些迹来刻画Mobius变换的几何。此外，利用表示L的富氏群G到特殊线性群SL(2，C)的升力，给出了黎曼曲面上的单圈S可除的充要条件。例如，如果S是亏格p(&gt；1)的紧黎曼曲面，则得到如下结果：G的升程数是2的2次P次方。设g是G中对应于L的元素，则L除当且仅当对G的任何提升，对应于g的矩阵总是有负迹。i在…中进一步引入了与标号黎曼曲面上测地线的交角对应的新的角度参数，从而得到了TeichMuller空间的整体实解析式和简明表示。证明了TeichMuller空间仅用角度参数来刻画，对于典型的(1，1)，(2，0)和(3，0)型TeichMuller空间的这种角度参数空间的分析是很容易的。角度参数对应于发电机的轴线与标记的富氏群的这些乘积之间的交角。我发现这些轴的排列具有高度的对称性。我研究了这样的Mobius变换几何、迹和角度参数之间的关系。通过这些观察，得到了角参数之间的许多关系和信息。其次，考虑了仅用角参数来表示Teichmuller模群的情况。重点研究了以下内容：1.将Teichmuller模群解释为某些特殊的以轴为界的双曲多边形间的相互作用；2.表示这些群的角度参数和长度参数之间的关系。较少