Angle parameters for Teichmüller spaces and their application
by
Yoshihide Okumura
Department of Mathematics, Faculty of Science, Shizuoka University, Japan

1 Introduction

Teichmüller space is the set of equivalence classes of marked Fuchsian groups. This space becomes a global real analytic manifold. Various parametrizations of these spaces are considered.

The purpose of this talk is to discuss global real analytic parametrizations of Teichmüller spaces and representations of Teichmüller modular groups (the mapping class groups).

2 Length parameters and angle parameters

Since R. Fricke's time, it is essentially known that Teichmüller space is parametrized global real analytically by some lengths of the closed geodesics on the Riemann surfaces represented by marked Fuchsian groups (see the works of R. Fricke-F. Klein, L. Keen and M. Seppälä-T. Sorvali, etc.). These lengths are called length parameters.

In the hyperbolic geometry, a triangle is determined by three sides or three interior angles. Hence we can deduce that Teichmüller space is parametrized global real analytically by some intersection angles between the geodesics on the Riemann surfaces represented by marked Fuchsian groups. We call these angles angle parameters.

In order to obtain global real analytic and simple representations of Teichmüller spaces, I introduced new angle parameters. I showed that Teichmüller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmüller spaces. We will show these angle parameter spaces.

3 Teichmüller modular groups

Considering the axes of the hyperbolic 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 Möbius transformations, traces and angle parameters, using the one-half powers of Möbius transformations and 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 Teichmüller modular groups by only angle parameters. We will consider the following:

• Relation between angle parameters and length parameters representing these groups.
• Description of such groups by only angle parameters.

Date received: June 11, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-02.