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I know that I can get a double torus $T^2$ from a regular $8$-gon in the hyperbolic plane by choosing appropriate side pairings. Likewise for a regular $10$-gon. I am told that I cannot always do this for a regular $p$-gon, where $p$ is arbitrarily large.

Could you help me see why not?

The polygons mentioned above, should all be fundamental domains of $\mathbb D$ (the Poincarré disk) and the map $f: (p\text{-gon})^\circ\to T^2$, where $(p\text{-gon})^\circ$ is the interior of the polygon, should be injective.

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Whatever polygon gluing diagram you already have, you can automatically increase the number of sides by $2$, like this: pick a new letter $x$ not already occurring, and add the prefix $x^{\vphantom{-1}} x^{-1}$ to the gluing word.

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  • $\begingroup$ I am unfamiliar with your terminology, it seems. What do you consider a gluing word? $\endgroup$ – gebruiker Mar 11 '17 at 12:56
  • $\begingroup$ The gluing word is a way to make precise the "appropriate side pairings". For example, I will guess that the side pairings you have in mind for obtaining the double torus have the form of the gluing word $aba^{-1}b^{-1}cdc^{-1}d^{-1}$. $\endgroup$ – Lee Mosher Mar 11 '17 at 13:50
  • $\begingroup$ It might help to look up a proof of the classification of surfaces; many elementary topology books cover this theorem. Gluing words are an important part of such proofs. $\endgroup$ – Lee Mosher Mar 11 '17 at 13:52

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