Hello Fractalus and welcome!
I'm not sure what your question is. You have an -gon in the hyperbolic plane, and you want to tile the hyperbolic plane with images of it under some group of transformations? So that your -gon becomes a fundamental region for the action of the group?
If I understand correctly, you then want to show that the quotient of the hyperbolic plane by the action of the group in question results in a Riemann surface of genus (and not gender ). Isn't this just a matter of identifying the sides of the fundamental region in an appropriate fashion?
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