# Von Dyck group

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This article defines a family of groups whose members are parametrized by tuples of natural numbers. In other words, for every tuple of natural numbers, there is a unique corresponding group (upto isomorphism) in that family

## Definition

### 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.

## The three types

### 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. Further information: Classification of finite subgroups of SO(3,R)

### 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.

## Particular cases

Smallest parameter Middle parameter Largest parameter Common name for group Group order Symmetry object
1 $n$ $n$ Cyclic group $n$ Regular polygon, symmetries in $SO(2,\R)$
2 2 $n$ Dihedral group $2n$ Regular polygon, symmetries in $O(2,\R)$ or in $SO(3,\R)$
2 3 3 Alternating group:A4 $12$ Regular tetrahedron, symmetries in $SO(3,\R)$
2 3 4 Symmetric group:S4 $24$ Cube or octahedron, symmetries in $SO(3,\R)$
2 3 5 Alternating group:A5 $60$ Icosahedron or dodecahedron, symmetries in $SO(3,\R)$