# triangle groups

Consider the following group presentation:

$$\mathrm{\Delta}(l,m,n)=\u27e8a,b,c:{a}^{2},{b}^{2},{c}^{2},{(ab)}^{l},{(bc)}^{n},{(ca)}^{m}\u27e9$$ |

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:

$$D(l,m,n)=\u27e8x,y:{x}^{l},{y}^{m},{(xy)}^{n}\u27e9$$ |

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*.

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 |