All januarials constructed from Hecke groups

TL;DR

This paper investigates the existence and construction of januarials derived from Hecke groups acting on finite projective lines, providing conditions, methods, and formulas for their genus calculation.

Contribution

It introduces a new method to find all januarials from Hecke groups acting on projective lines and derives a formula for their genus based on fixed points.

Findings

Established conditions for januarials from Hecke groups

Developed a comprehensive method for constructing januarials

Derived a formula to calculate the genus of januarials

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.