### Abstract

Here we show that if G is an arithmetic Fuchsian group of genus 0, then the totally real defining field k of G must be such that [k : Q] = 11. The same inequality holds for discrete arithmetic hyperbolic reflection groups acting on a two-dimensional hyperbolic space H2. In addition, we show that there exists an arithmetic Fuchsian group of genus 0 containing an element of order N if and only if N ¿ {2, 3, …, 16, 18, 20, 22, 24, 26, 28, 30, 36}. A slightly less precise statement holds for discrete arithmetic hyperbolic reflection groups acting on H2.

Original language | English |
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Pages (from-to) | 111-123 |

Number of pages | 13 |

Journal | Bulletin of the London Mathematical Society |

Volume | 43 |

Issue number | 1 |

Early online date | 2 Nov 2010 |

DOIs | |

Publication status | Published - Feb 2011 |

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## Cite this

MacLachlan, C. (2011). Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2.

*Bulletin of the London Mathematical Society*,*43*(1), 111-123. https://doi.org/10.1112/blms/bdq085