### Abstract

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 |
---|---|

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

*Pure and Applied Mathematical Quarterly*,

*2*(2), 569-599.

**Arithmetic Fuchsian groups of genus zero.** / Long, Darren; MacLachlan, Colin; Reid, Alan.

Research output: Contribution to journal › Article

*Pure and Applied Mathematical Quarterly*, vol. 2, no. 2, pp. 569-599.

}

TY - JOUR

T1 - Arithmetic Fuchsian groups of genus zero

AU - Long, Darren

AU - MacLachlan, Colin

AU - Reid, Alan

PY - 2006/4

Y1 - 2006/4

N2 - 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.

AB - 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.

KW - rational homology

KW - modular group

KW - fields

KW - subgroups

KW - curves

KW - forms

M3 - Article

VL - 2

SP - 569

EP - 599

JO - Pure and Applied Mathematical Quarterly

JF - Pure and Applied Mathematical Quarterly

SN - 1558-8599

IS - 2

ER -