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Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 with $(2,3,7)$ tiling on it.

Is there any way to see explicitly what will be a fundamental domain of $\Gamma$ inside the $(2,3,7)$ tiling of the plane?

The fundamental domain must have the following properties:

  1. It will contain 168 $(2,3,7)$ triangle.
  2. If we take the presentation of $\Gamma$ is $\langle g_1,g_2,g_3,g_4 \mid [g_1,g_2][g_3,g_4] \rangle$ then the fundamental domain will be an hyperbolic octagon.
  3. Alternate edges of that octagon has same length.

Will it also be true that the sum of all internal angles of the octagon is $2\pi$?

I want to see a fundamental domain explicitly as part of the tiling.

Thanks in advance.

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  • $\begingroup$ A trivial remark - if the fundamental domain is a regular (hyperbolic) octagon, it cannot be assembled from the whole (2,3,7)-triangles since angles of the octagon are $\pi/4$ and you cannot combine $\pi/2$, $\pi/3$ and $\pi/7$ angles into a $\pi/4$ $\endgroup$ Aug 30, 2022 at 6:39
  • $\begingroup$ Yes. The octagon need not be regular. $\endgroup$
    – KAK
    Aug 30, 2022 at 6:40
  • $\begingroup$ Let me add that answers to the question regular tiling of a surface of genus 2 by heptagons might help $\endgroup$ Aug 30, 2022 at 6:46
  • $\begingroup$ In fact I believe the (2,3,7)-triangle group does not have any torsion free genus 2 subgroups. In the paper Geometric uniformization in genus 2 (T. Kuusalo and M. Näätänen, Ann. Acad. Sci. Fenn. Ser. A. I. 20, 1995, 401-418) it is proved that the only triangle groups having such subgroups are (2, 3, 8), (2, 4, 6), (2, 4, 8), (2, 5, 10), (2, 6, 6), (2, 8, 8), (3, 3, 4), (3, 4, 4), (3, 6, 6), (4, 4, 4) and (5, 5, 5). $\endgroup$ Aug 30, 2022 at 6:57
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    $\begingroup$ No, it does not. Actually irregular such tilings are known. One of the answers to the question I linked contains a link to a picture - unfortunately with four holes left unglued $\endgroup$ Aug 30, 2022 at 10:20

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