All januarials constructed from Hecke groups

arXiv1810.00203MaRDI QIDQ6307474

Saadia Mehwish, Qaiser Mushtaq

Publication date: 29 September 2018

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 $H_{Lambd{a}_{ell}},$ when acting on the projective lines over finite fields $PL(F_q)$. We develope a method to find all the januarials from Hecke groups $H_{Lambd{a}_{ell}}$, 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.

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