THE BERS EMBEDDING OF ONE-DIMENSIONAL TEICHMÜLLER SPACES

TOSHIYUKI SUGAWA

This is joint work with Yohei Komori, and partially with Masaaki Wada and Yasushi Yamashita.

In this poster, we explain how to compute the monodromy representation of a Fuchsian group associated to a given (bounded) projective structure on a marked once-punctured torus or four-times punctured sphere, say X. The calculation can essentially be reduced to solving (numerically) the homogeneous linear ordinary differential equation

2y′′ + \{\frac{1}{2z^{2}(z − 1)^{2}} + \frac{1}{2(z − λ)^{2}} + \frac{t + c(λ)}{z(z − 1)(z − λ)}\} y = 0,

where λ is a point in ℂ \ {0, 1} such that X and the domain Y = ℂ \ {0, 1, λ} are commensurable with respect to coverings, and c(λ) is a constant which is often called the accessary parameter of Y. (This method is a modification of that of R. M. Porter and L. Keen developed around 1980.)

Using this, we exhibit pictures of the Bers embedded Teichmüller space of X, and furthermore, pictures of the interior of the discreteness locus in the space of bounded projective structures on X. Note that the component of the iterior of the discreteness locus containing the “origin” is nothing but the (Bers embedded) Teichmüller space of X by a result of H. Shiga. A component of the interior of the discreteness locus other than the Teichmüller space of X is called exotic. We will see a mysterious configuration of the exotic components.

We also provide computer graphics of the bending loci corresponding to simple closed curves on X. As applications, numerically, we can calculate the accessary parameter, and we can give the inner (and possibly, outer) radius of the Bers embedding of the Teichmüller space of the surface ℂ \ {0, 1, −1}. Also, we can give the Maskit coordinate of a given Teichmüller parameter in a numerical way.

Information: You can see related pictures in the following website: http://www.kusm.kyoto-u.ac.jp/complex/Bers/

Department of Mathematics, University of Helsinki,

P. O. Box 4 (Yliopistonkatu 5) FIN-00014 Helsinki, Finland

  E-mail address: sugawa@helsinki.fi

1991 Mathematics Subject Classification. Primary 30F60; Secondary 30F40, 30-04.

Key words and phrases. Teichmüller space, Bers embedding, monodromy, pleating locus, bending coordinate, once-punctured torus .