Hyperbolic construction

The (2,3,7) triangle group is the group of orientation-preserving isometries of the tiling by the (2,3,7) Schwarz triangle, shown here in a Poincaré disk model projection.

To construct the triangle group, start with a hyperbolic triangle with angles π/2, π/3, and π/7. This triangle, the smallest hyperbolic Schwarz triangle, tiles the plane by reflections in its sides. Consider then the group generated by reflections in the sides of the triangle, which (since the triangle tiles) is a non-Euclidean crystallographic group (a discrete subgroup of hyperbolic isometries) with this triangle for fundamental domain; the associated tiling is the order-3 bisected heptagonal tiling. The (2,3,7) triangle group is defined as the index 2 subgroups consisting of the orientation-preserving isometries, which is a Fuchsian group (orientation-preserving NEC group).

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732)							[7,3]+, (732)
{7,3}	t{7,3}	r{7,3}	t{3,7}	{3,7}	rr{7,3}	tr{7,3}	sr{7,3}
Uniform duals
V73	V3.14.14	V3.7.3.7	V6.6.7	V37	V3.4.7.4	V4.6.14	V3.3.3.3.7

Group presentation

It has a presentation in terms of a pair of generators, g₂, g₃, modulo the following relations:

${\displaystyle g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=1.}$

Geometrically, these correspond to rotations by ${\displaystyle {\frac {2\pi }{2}},{\frac {2\pi }{3}}}$, and ${\displaystyle {\frac {2\pi }{7}}}$ about the vertices of the Schwarz triangle.

Quaternion algebra

The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1.

Let η = 2cos(2π/7). Then from the identity

${\displaystyle (2-\eta )^{3}=7(\eta -1)^{2}.}$

we see that Q(η) is a totally real cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i² = j² = η, ij = −ji. One chooses a suitable Hurwitz quaternion order ${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}$ in the quaternion algebra. Here the order ${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}$ is generated by elements

${\displaystyle g_{2}={\tfrac {1}{\eta }}ij}$

${\displaystyle g_{3}={\tfrac {1}{2}}(1+(\eta ^{2}-2)j+(3-\eta ^{2})ij).}$

In fact, the order is a free Z[η]-module over the basis ${\displaystyle 1,g_{2},g_{3},g_{2}g_{3}}$. Here the generators satisfy the relations

${\displaystyle g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=-1,\,}$

which descend to the appropriate relations in the triangle group, after quotienting by the center.