The Farey maps modulo n

Abstract

Let $U^{\star}$ denote the upper-half plane compactified by adding the points$Q \cup {\infty}$ to the upper half plane $U$ . On $U$ we have the universaltriangular map $M_3$ which can be realised by the well-known Fareymap as described below. These have as vertices the extended rationals$Q \cup {\infty}$. Our aim in this paper is to discuss the maps (or cleandessin d’enfants) $M_3=/\Gamma(n)$ which lies on the Riemann surface $U^{\star}/\Gamma (n)$where $\Gamma (n)$ is the principal congruence subgroup mod n of the classicalmodular group $\Gamma$. These have vertices as rational numbers "modulo$n$". These were introduced in [4] and also discussed in [8].