triangle groups

Consider the following group presentation:


where l,m,n.

A group with this presentationMathworldPlanetmathPlanetmath corresponds to a triangle; roughly, the generatorsPlanetmathPlanetmathPlanetmath are reflections in its sides and its angles are π/l,π/m,π/n.

Denote by D(l,m,n) the subgroupMathworldPlanetmathPlanetmath of index ( 2 in Δ(l,m,n), corresponding to preservation of of the triangle.

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


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

Arising from the geometrical nature of these groups,


is called the spherical case,


is called the Euclidean case, and


is called the hyperbolic case

Groups either of the form Δ(l,m,n) or D(l,m,n) are referred to as triangle groups; groups of the form D(l,m,n) are sometimes refered to as von Dyck groups.

Title triangle groups
Canonical name TriangleGroups
Date of creation 2013-03-22 14:25:07
Last modified on 2013-03-22 14:25:07
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Definition
Classification msc 20F05
Related topic ExamplesOfGroups
Defines von Dyck groups