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Ref: Tiling the hyperbolic plane by non-regular quadrilaterals

Question: What is known about the tilings of the hyperbolic plane by n-gons that are not regular, especially for values of n greater than 6?

Or is it that for (say) n > some finite value, maybe even 6, the only n-gons that can tile the hyperbolic plane are regular (equal angles and sides)ones?

Note: In above referenced post, one had suggested a weaker claim:
“If it can be shown (say) that "for large enough n, any n-gon that tiles the hyperbolic plane has necessarily to have all angles except at most, a constant number (or very small fraction of n) of angles equal" that could be hopefully interesting.”

Note added after answer: I did mean only convex n-gons although it wasn’t specified above.

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Nothing like this is true. Let $D_0$ be a fundamental polygon of a Fuchsian group, then it has a side pairing $s\to s'$ so that for each pair the generator $f$ of the group sends $s\to s'$. Now, if we replace the hyperbolic geodesic segment $s$ by a hyperbolic broken line, $s_1$ close to $s$ and replace $s'$ by $s_1'=f(s_1)$, we obtain a polygon with more sides and more angles, which is fundamental for the same group.

Of course, you can add a condition that your polygons are convex, but this also does not help.

The general philosophical reason is that regular polygons for fixed $n$ make a $1$-parametric family (you can take the angle or the side length as parameter), while convex polygons must serve as fundamental regions for all Fuchsian groups parametrizing compact Riemann surfaces, and there is a $6g-6$-parametric family of such surfaces, where $g$ is the genus. They correspond to convex polygons with $4g$ vertices.

On your first question "what is known?" the main result is called the Poincare theorem on polygons which in some sense describes all polygons which tile the hyperbolic plane as fundamental regions of Fuchsian groups. See, for example

B. Maskit, Kleinian groups, Springer 1980, section IV.H-I.

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  • $\begingroup$ Thanks. Hope u could also clarify a query made in this earlier post: mathoverflow.net/questions/398191/…: which triangle is worst for tiling the hyperbolic plane? $\endgroup$ Commented Aug 10 at 15:53

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