Arithmetic and geometry of the Hecke groups

We study the arithmetic and geometry properties of the Hecke group Gq. In particular, we prove that Gq has a subgroup X of index d, genus g with v∞ cusps, and τ2 (resp. vri) conjugacy classes of elliptic elements that are conjugates of S (resp. Rq/ri) if and only if (i) 2g−2+τ2/2+∑i=1kvri(1−1/ri)+v∞=d(1/2−1/q), and (ii) m0=4g−4+τ2+2v∞+∑i=1kvri(2−q/ri)≥0 is a multiple of q−2. Note that if q is odd (resp. prime), then m0/(q−2)∈Z (resp. N∪{0}) is a consequence of (i).