All januarials constructed from Hecke groups

Abstract

Professor Graham Higman defined januarial as a special instance of map constructed from embedding of a coset diagram for an action of $\Delta (2,\ell ,k)$, on finite sets yielding exactly two orbits of the product of the two generators, having equal sizes. In this paper we determine a condition for the existence of a januarial from $\Delta (2,\ell ,k),$ the quotients of Hecke groups $H_{\Lambda _{\ell }},$ when acting on the projective lines over finite fields $PL(F_{q})$. We develope a method to find all the januarials from Hecke groups $H_{\Lambda _{\ell }}$, when the triangle group $\Delta (2,\ell ,k)$ acts on $PL(F_{q})$. We evelove a formula for calculating genus of coset diagram depending on the fixed points. By using it, we determine genus of the januarials.