From Poincare's Theorem, the side-pairing transformations for a fundamental polygon will generate a Fuchsian group. Existentially and constructively speaking, how am I going to construct or look for a set of side-pairing transformations that will eventually form a Fuchsian group? Are there any initial qualifications for these isometries?
I am 'moving' within the Disk model and the isometric circles are my lead. With these, I am trying to construct side-pairing transformations for the possible configurations of an octagon that will eventually form a 2-hole torus.