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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Some presentations for $\overline{\Gamma}_0(N)$


Author: Antonio Lascurain Orive
Journal: Conform. Geom. Dyn. 6 (2002), 33-60
MSC (2000): Primary 11F06, 20H05, 30F35, 51M10, 52C22; Secondary 13M05, 22E40
Published electronically: May 30, 2002
MathSciNet review: 1948848
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Abstract: Some presentations of the Fuchsian groups defined by the Hecke congruence subgroups

\begin{displaymath}\Gamma_{0}( N)\;=\; \left\{\begin{pmatrix} a& b \\ c& d \en... ...} )\;\Big {\vert} \;\; c\equiv 0\;\; \text{mod}\; N \right\} \end{displaymath}

are given. The first is one obtained by the Reidemeister-Schreier rewriting process, thereby completing and correcting Chuman's work on the subject. The main result (Theorem 3) is the reduction of this huge presentation into another one which is simple and useful. In the process, $\mathbb{Z} _N$ is partitioned into three subsets that exhibit many cyclic and dual properties of its ring structure. For some cases, a minimal presentation derived from the Ford domains is given explicitly in terms of the units and its inverses.


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Additional Information

Antonio Lascurain Orive
Affiliation: Havre 101, Colonia Villa Verdun, Mexico D.F. 01810 Mexico
Email: lasc@hp.fciencias.unam.mx

DOI: http://dx.doi.org/10.1090/S1088-4173-02-00073-5
PII: S 1088-4173(02)00073-5
Received by editor(s): January 8, 2001
Received by editor(s) in revised form: April 11, 2002
Published electronically: May 30, 2002
Article copyright: © Copyright 2002 American Mathematical Society