Triangle Group - Von Dyck Groups

Von Dyck Groups

Denote by D(l,m,n) the subgroup of index 2 in Δ(l,m,n) generated by words of even length in the generators. Such subgroups are sometimes referred to as "ordinary" triangle groups or von Dyck groups, after Walther von Dyck. For spherical, Euclidean, and hyperbolic triangles, these correspond to the elements of the group that preserve the orientation of the triangle – the group of rotations. For projective (elliptic) triangles, they cannot be so interpreted, as the projective plane is non-orientable, so there is no notion of "orientation-preserving". The reflections are however locally orientation-reversing (and every manifold is locally orientable, because locally Euclidean): they fix a line and at each point in the line are a reflection across the line.

The groups D(l,m,n) is defined by the following presentation:

In terms of the generators above, these are x = ab, y = ca, yx = cb. Geometrically, the three elements x, y, xy correspond to rotations by 2π/l, 2π/m and 2π/n about the three vertices of the triangle.

Note that D(l,m,n) ≅ D(m,l,n) ≅ D(n,m,l), so D(l,m,n) is independent of the order of the l,m,n.

A hyperbolic von Dyck group is a Fuchsian group, a discrete group consisting of orientation-preserving isometries of the hyperbolic plane.