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.
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.