triangle groups

Consider the following group presentation:

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

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 $\mathrm{\Delta }(l,m,n)$, corresponding to preservation of of the triangle.

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

$$D(l,m,n)=⟨x,y:x^l,y^m,{(xy)}^n⟩$$

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

is called the hyperbolic case

Groups either of the form $\mathrm{\Delta }(l,m,n)$ 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.