(2,3,7) triangle group

Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.

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.

It has a presentation in terms of a pair of generators, g2, g3, modulo the following relations: Geometrically, these correspond to rotations by

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.

Then from the identity 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.

is generated by elements In fact, the order is a free Z[η]-module over the basis

Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices.

Thus, traces of group elements (and hence also translation lengths of hyperbolic elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups) can be calculated by means of the reduced trace in the quaternion algebra, and the formula