### Abstract

If ¡ is a finite co-area Fuchsian group acting on H2, then the quotient H2=¡ is a

hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with

punctures) and a finite number of cone points. Through their close connections

with number theory and the theory of automorphic forms, arithmetic Fuchsian

groups form a widely studied and interesting subclass of finite co-area Fuchsian

groups. This paper is concerned with the distribution of arithmetic Fuchsian

groups ¡ for which the underlying surface of the orbifold H2=¡ is of genus zero;

for short we say ¡ is of genus zero.

hyperbolic 2-orbifold, with underlying space an orientable surface (possibly with

punctures) and a finite number of cone points. Through their close connections

with number theory and the theory of automorphic forms, arithmetic Fuchsian

groups form a widely studied and interesting subclass of finite co-area Fuchsian

groups. This paper is concerned with the distribution of arithmetic Fuchsian

groups ¡ for which the underlying surface of the orbifold H2=¡ is of genus zero;

for short we say ¡ is of genus zero.

Original language | English |
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Pages (from-to) | 569-599 |

Journal | Pure and Applied Mathematical Quarterly |

Volume | 2 |

Issue number | 2 |

Publication status | Published - Apr 2006 |

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### Keywords

- rational homology
- modular group
- fields
- subgroups
- curves
- forms

### Cite this

Long, D., MacLachlan, C., & Reid, A. (2006). Arithmetic Fuchsian groups of genus zero.

*Pure and Applied Mathematical Quarterly*,*2*(2), 569-599.