Von Dyck group

Definition in terms of presentation
For natural numbers $$l,m,n$$, the von Dyck group $$D(l,m,n)$$ is defined by the following presentation:

$$\langle a,b,c|a^l = b^m = c^n = abc = e \rangle$$

where $$e$$ denotes the identity element.

This is a subgroup of index two in the triangle group, but some people use the term triangle group for the von Dyck group.

Geometric description
Given natural numbers $$l,m,n$$, consider a triangle with sides $$\pi/l, \pi/m, \pi/n$$ on a suitable simply connected Riemannian surface (i.e., a suitable model for Euclidean or non-Euclidean geometry). The von Dyck group is the group generated by rotations about the vertices of the triangle by angles of $$2\pi/l$$, $$2\pi/m$$, $$2\pi/n$$ respectively.

Spherical von Dyck groups
The triple $$(l,m,n)$$ in this case satisfies:

$$\frac{1}{l} + \frac{1}{m} + \frac{1}{n} > 1$$,

The solutions to which are $$(2,3,3), (2,3,4), (2,3,5)$$, and $$(2,2,n)$$.

This is the spherical case, with the model being the unit sphere in three-dimensional space, and the corresponding von Dyck groups are termed spherical von Dyck groups. Spherical von Dyck groups are subgroups of the special orthogonal group $$SO(3,\R)$$, because $$SO(3,\R)$$ is precisely the group of orientation-preserving isometries of the sphere. All of these turn out to be finite subgroups of $$SO(3,\R)$$, and these also turn out to be the only finite subgroups of $$SO(3,\R)$$, a fact that follows from Euler's theorem and some additional work. The finiteness can also be viewed as a consequence of the fact that the sphere is compact and simply connected.

Euclidean von Dyck groups
The triple $$(l,m,n)$$ in this case satisfies:

$$\frac{1}{l} + \frac{1}{m} + \frac{1}{n} = 1$$,

for which the only solutions are $$(4,4,2)$$ and $$(3,3,3)$$, i.e., the right isosceles triangle and the equilateral triangle in the usual Euclidean plane.

Both of these give wallpaper groups, and neither is finite.

Hyperbolic von Dyck groups
$$\frac{1}{l} + \frac{1}{m} + \frac{1}{n} < 1$$,

for which there are infinitely many solutions. The model for this is the hyperbolic plane.