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

Abstract:

Some presentations of the Fuchsian groups defined by the Hecke congruence subgroups \[ \Gamma _{0}( N)\;=\; \left \{\begin {pmatrix} a& b c& d \end {pmatrix} \in SL(2,\mathbb {Z})\;\Big {|} \;\; c\equiv 0\;\; \text {mod}\; N \right \} \] 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.