Fundamental Polygon - Explicit Form For Standard Polygons

Explicit Form For Standard Polygons

Explicit expressions can be given for the standard polygons. One of the more useful forms is in terms of the group associated with the standard polygon. For a genus oriented surface, the group may be given by generators . These generators are given by the following fractional linear transforms acting on the upper half-plane:

a_k= \left( \begin{matrix} \cos k\alpha & -\sin k\alpha \\ \sin k\alpha & \cos k\alpha \end{matrix} \right) \left( \begin{matrix} e^p & 0 \\ 0 & e^{-p} \end{matrix} \right) \left( \begin{matrix} \cos k\alpha & \sin k\alpha \\ -\sin k\alpha & \cos k\alpha \end{matrix} \right)

for . The parameters are given by

and

and

It may be verified that these generators obey the constraint

which gives the totality of the group presentation.

Read more about this topic:  Fundamental Polygon

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