# triangle groups

Consider the following group presentation:

 $\Delta(l,m,n)=\langle a,b,c:a^{2},b^{2},c^{2},(ab)^{l},(bc)^{n},(ca)^{m}\rangle$

where $l,m,n\in\mathbb{N}$.

A group with this presentation corresponds to a triangle; roughly, the generators are reflections in its sides and its angles are $\pi/l,\pi/m,\pi/n$.

Denote by $D(l,m,n)$ the subgroup of index (http://planetmath.org/Coset) 2 in $\Delta(l,m,n)$, corresponding to preservation of of the triangle.

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

 $D(l,m,n)=\langle x,y:x^{l},y^{m},(xy)^{n}\rangle$

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

Arising from the geometrical nature of these groups,

 $1/l+1/m+1/n>1$

is called the spherical case,

 $1/l+1/m+1/n=1$

is called the Euclidean case, and

 $1/l+1/m+1/n<1$

is called the hyperbolic case

Groups either of the form $\Delta(l,m,n)$ 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 TriangleGroups 2013-03-22 14:25:07 2013-03-22 14:25:07 rmilson (146) rmilson (146) 10 rmilson (146) Definition msc 20F05 ExamplesOfGroups von Dyck groups