Analysis and Geometry of the Teichmuller Spaces

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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