Principal congruence subgroups of the Hecke groups (Q5928006)

The authors apply L. E. Dickson's determination of the subgroups of \(\text{PSL}(2, {\mathbb{F}}_q)\) to find invariants of congruence subgroups of the Hecke triangle Fuchsian groups, NEWLINE\[NEWLINEG_n :=\left\langle \begin{pmatrix} 0 &-1\\1 &0 \end{pmatrix}, \begin{pmatrix} 1 &2 \cos \pi/n\\0 &1 \end{pmatrix} \right\rangle .NEWLINE\]NEWLINE The authors' main theorem is in fact a direct implication of Satz 3.2 (p. 78) in \textit{W. Borho}'s [Abh. Math. Sem. Univ. Hambg. 39, 76-82 (1973; Zbl 0267.10034)]. The applications to geometric invariants of the specific congruence subgroups are new.