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Defining Riemann surfaces

Enviado por Mika.Seppala el Lun, 2008/02/11 - 23:04.

Excluding certain special cases, a Riemann surface $ X $ has the unit disk $ D $ as its universal cover. Hence

\[X=D/G\]

where $ G $ is a group of Möbius transformations. If $ X $ is compact, $ G $ is finitely generated.

When defining Riemann surfaces for computations one may simply give the generators of the group $ G $ (and the relations that the generator satisfy).

Now the problem is:

  • How to find Möbius transformations which generate a discontinuous group?
  • How to perform the Fenchel Nielsen twists and other deformations of such group?

Suitable OpenMath CDs also need to be produced for these computations.

I hope that this forum will serve to coordinate the efforts at EPFL, U of Helsinki and at the Florida State University in this matter.

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Riemann surfaces

Enviado por Buser, Peter el Mar, 2008/02/12 - 01:11.

I'll have a number of questions concerning the format for the description of a Riemann surface using uniformizing fuchsian groups. I'll start with these tomorrow.
Peter

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current format

Enviado por Lebedev, Yuri el Vie, 2008/02/15 - 01:59.

Hi Dr. Buser
We represent mobius transformations in the following format:

config{
[0]Mob(HLine(HPoint(x1,y1),HPoint(x2,y2),k1);
[1]Mob(HLine(HPoint(x3,y3),HPoint(x4,y4),k2);
[2]Mob(HLine(HPoint(x5,y5),HPoint(x6,y6),k3);
.
.
[j]Mob(HLine(HPoint(x2j-1,y2j-1),HPoint(x2j,y2j),kj);
.
.
Word([0],[1],[2],inv[2],inv[0],inv[j],inv[0],inv[1]);
Word([0],inv[1],[1],inv[0]);
Word([0],[1],inv[j],inv[0],..........,inv[k],.....);
.
.
.
}

Where each mobius transformation is represented by attracting,repelling points and a multiplier.
The above format gets parsed into programs internal representation. Also, once it is in the aboves format it can be easily reparsed to be represented by OpenMath using 2 proposed CDs: HypGeo1 and Mobius1 to be sent across the wire using TCP/IP java sockets.

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