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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.

Hades Pluto
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  • Poincare's polygon theorem is more specific than you have stated: there are explicit hypotheses that the polygon and its side pairing transformation must satisfy. Do you know those hypotheses? – Lee Mosher Jan 21 '21 at 15:41
  • Lifting the following statements from one of my references, these are the assumptions: "Let D be a convex hyperbolic polygon with finitely many sides. Suppose that all vertices lie inside U and that D is equipped with a collection of side-pairing Mobius transformations. Suppose that no side of D is paired with itself. Suppose that the elliptic cycles are E1; : : : ; Er, and that each Ej is a proper elliptic cycle with corresponding cycle constant mj ." One of the consequences is that the side-pairing transformations will generate a Fuchsian group. – Hades Pluto Jan 22 '21 at 03:56
  • So your question seems to be what, exactly, are side-pairing transformations? – Lee Mosher Jan 22 '21 at 03:57
  • Based on this version, as I see it, there are already side-pairing transformations for the polygon. What I am looking for are how do these side-pairing transformations be qualified to form an octagon? – Hades Pluto Jan 22 '21 at 04:00
  • I would like to be enlightened, what are the initial qualifications for the side-pairing transformations so that, together with the other side-pairing transformations, eventually helpful to form an octagon? – Hades Pluto Jan 22 '21 at 04:02
  • I still don't really understand what you are asking. In the statement of the Poincare polygon theorem, the polygon AND it's side pairing transformations are both given. You don't "form an octagon" out of side pairing transformations. Instead, you form a surface out of the polygon and its side pairing transformations (by forming a quotient space of an appropriate kind). – Lee Mosher Jan 22 '21 at 04:04
  • What might make sense to ask is: How does one construct an octagon and side pairing transformations on that octagon, so that the surface (obtained as a quotient by applying the Poincare polygon theorem to that octagon and those side pairing transformations) is a closed surface of genus 2? So... is that kind of what you are asking? – Lee Mosher Jan 22 '21 at 04:07
  • Let me put it this way. Does that mean that any hyperbolic isometry can become a side-pairing transformation for some octagons? for any octagons? – Hades Pluto Jan 22 '21 at 04:09
  • Wow. Your rewording is indeed outstanding. Thank you. I will be using it. – Hades Pluto Jan 22 '21 at 04:10
  • Let me think about an answer (and others might as well), but meanwhile it would help to edit your question to reflect the clarifications we have been discussing. – Lee Mosher Jan 22 '21 at 14:40

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