Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$

Abstract

In this paper, we investigate suborbital graphs $G_{u,n}$ of the normalizer $\Gamma_B(N)$ of $\Gamma_0(N)$ in $PSL(2,\mathbb{R})$ for $N= 2^\alpha 3^\beta$, where $\alpha=0,2,4,6$ and $\beta =1,3$. In each of these cases, the normalizer becomes a triangle group and the graph arising from the action of the normalizer contains hexagonal circuits. In order to obtain graphs, we first define an imprimitive action of $\Gamma _B(N)$ on $\widehat{\mathbb{Q}}$ using the group $H_B(N)$ and then we obtain some properties of the graphs arising from this action.