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Hometriangle groups

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# 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 2 in $\Delta(l,m,n)$, corresponding to preservation of orientation 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 independent of the order 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*.

## Mathematics Subject Classification

20F05*no label found*

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