We study the arithmetic and geometry properties of the Hecke group $G_q$. In particular, we prove that $G_q$ has a subgroup $X $ of index $d$, genus $g$ with $v_{ \infty} $ cusps, and $\tau_2$ (resp. $v_{ r_i}$) conjugacy classes of elements that are conjugates of $S$ (resp. $R^{ q/r_i}$) if and only if (i) $ 2g-2 + \tau_2/2 +\sum_{ i=1}^k v_{ r_i}(1-1/r_i) + v_{ \infty} = d(1/2-1/q)$, and (ii) $ m _0= 4g-4 +\tau_2 + 2 v_{ \infty} + \sum _{ i=1}^k v_{ r_i}(2-q/r_i)\ge 0$ is a multiple of $q-2$, (iii) $m \ge 0$. In the case $q$ is odd, (ii) is a consequence of (i).
Categories: math.GR