Triangle Group

Definition

Define the triangle group $T_{m,n,l}$ as $$ X=\langle x,y\big||x|=m,|y|=n,|xy|=l\rangle.

Consider $A:=(\frac{1}{m}+\frac{1}{n}+\frac{1}{l})\pi$.

Assume that $m⩽n⩽l$ and $G$ is nontrivial.

Case 1. $A>\pi$.

If $m=1$, then $X\cong {\mathbb{Z}}_n$.

If $m=2$ and $n=2$, then $X\cong D_{2l}$.

If $m=2$ and $n⩾3$, then $n=3$ and $l\in \{3,4,5\}$, which corresponding $A_4,S_4,A_5$, respectively.

Case 2. $A=\pi$

All possible solution: $(3,3,3),(2,3,6),(2,4,4)$. They correspond regular triangle, hexagon, square. They are the only polygons filling the whole plane.

Those groups are infinity and solvable.

Case 3. $A<\pi$

These groups are infinity and non-solvable.