(2,3,7) triangle group

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.
The term " triangle group" most often refers not to the full triangle group Δ, but rather to the ordinary triangle group D of orientation-preserving maps, which is index 2.
Torsion-free normal subgroups of the triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.

Constructions

Hyperbolic construction

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 is a non-Euclidean crystallographic group with this triangle for fundamental domain; the associated tiling is the order-3 bisected heptagonal tiling. The triangle group is defined as the index 2 subgroups consisting of the orientation-preserving isometries, which is a Fuchsian group.

Group presentation

It has a presentation in terms of a pair of generators, g₂, g₃, modulo the following relations:
Geometrically, these correspond to rotations by, and about the vertices of the Schwarz triangle.

Quaternion algebra

The 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. Then from the identity
we see that Q is a totally real cubic extension of Q. The 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 i''² = j² = η, ij = −ji. One chooses a suitable Hurwitz quaternion order in the quaternion algebra. Here the order is generated by elements
In fact, the order is a free Z-module over the basis. Here the generators satisfy the relations
which descend to the appropriate relations in the triangle group, after quotienting by the center.

Relation to SL(2,R)

Extending the scalars from Q to R, one obtains an isomorphism between the quaternion algebra and the algebra M of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the associated tilings, as depicted at right: the tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane.
For many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements can be calculated by means of the reduced trace in the quaternion algebra, and the formula