Arithmetic and geometry of the Hecke groups

Abstract

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) $m_0=4g-4+{\tau }_2+2v_{\infty }+{\sum }_{i=1}^k{v}_{r_i}(2-q/r_i)\ge 0$ is a multiple of $q-2$, (iii) $m\ge 0$. In the case $q$ is odd, (ii) is a consequence of (i).