Tiling of Klein quartic with 56 equilateral triangles I am interested in the regular hyperbolic 14-gon associated with the Klein quartic (Klein's famous "Hauptfigur"). This 14-gon, or more precisely the genus-3 surface obtained by identifying its edges in appropriate ways, can be tiled by 56 equilateral $2\pi/7$ triangles, as beautifully illustrated by Greg Egan here:
https://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html
On the other hand, I know the Klein 14-gon can be tiled by 336 Schwarz triangles (the smallest triangles in Egan's picture). This tiling is described by the (2,3,7) triangle group $\Delta(2,3,7)$:
$\Delta(2,3,7)=\langle a,b,c\mid a^2=b^2=c^2=(ab)^2=(bc)^3=(ca)^7=1\rangle$,
where $a,b,c$ are reflections across the 3 edges of a given Schwarz triangle. The edge identifications of the 14-gon are described by a torsion-free index-336 normal subgroup $\Gamma\triangleleft \Delta(2,3,7)$. The factor group $G=\Delta(2,3,7)/\Gamma$ is the order-336 group of automorphisms of the Klein surface (here including both conformal and anti-conformal automorphisms). Elements of $G$ "translate'' one Schwarz triangle to another, and thus tile the entire surface starting from any given Schwarz triangle.
Now, observing that each $2\pi/7$ triangle consists of 6 Schwarz triangles, here is what I want to know:

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*Analogously to $\Delta(2,3,7)$, is there an infinite (Fuchsian) group $\Delta'$ which tiles the hyperbolic plane with $2\pi/7$ triangles? If so, can $\Delta'$ be expressed as a (normal?) index-6 subgroup of $\Delta(2,3,7)$? What are its generators/relators in terms of $a,b,c$, assuming it is finitely generated/presented?


*Assuming $\Delta'$ above exists, is $\Gamma$ realized as a torsion-free index-56 normal subgroup of $\Delta'$?


*Correspondingly, is there a finite group $G'$ of order 56 which tiles the Klein 14-gon with $2\pi/7$ triangles? If so, can $G'$ be expressed as a (normal?) index-6 subgroup of $G$?
 A: No, there is no Fuchsian group $\Delta'$ whose fundamental domains are $2\pi/7$ equilateral triangles.
To see this, consider the tiling using these large triangles, and divide each large triangle into six small triangles as in $\Delta(2,3,7)$. The group $\Delta'$ acts on the subtriangles with exactly six orbits. Now consider some vertex $V$ of a large triangle, and consider the seven positively oriented small triangles incident to $V$. By the pigeonhole principle two of these must be in the same $\Delta'$-orbit, which implies that the rotation around $V$ mapping one to the other is in $\Delta'$. A multiple of this rotation must be a rotation by $2\pi/7$, hence this rotation by $2\pi/7$ is in $\Delta'$.
However, the same argument applies to any other vertex of any large triangle, including a vertex $W$ neighboring $V$. Rotating by $2\pi/7$ around $W$ then rotating by $2\pi/7$ around $V$ yields a rotation by $2\pi/3$ around the center of one of the large triangles incident to $V$ and $W$, this rotation is by construction an element of $\Delta'$. This contradicts the assumption that the large triangle is a fundamental domain of $\Delta'$.
