Normalizer Maps Modulo N

Abstract

The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ of ${\mathsf{\Gamma }}_0\left(N\right)$ modulo N, where N denotes a positive integer. We reveal the complete structure of these maps, finding their vertices, edges, darts, and faces explicitly. The maps we investigated in the present paper were all regular maps of large genus except for some low values of N.

1. Introduction

We will denote the group of all linear fractional transformations by $PSL(2,\mathbb{R})$. By a linear fractional transformation we mean a function of a complex variable z, defined by

$$\begin{array}{c}\hfill T:z\to \frac{az+b}{cz+d},\mathrm{where}a,b,c\mathrm{and}d\mathrm{are} \mathbb{R} \mathrm{and}ad-bc=1.\end{array}$$

It is well-known that the linear fractional transformations can be represented by matrices. Hence, we can regard the elements of $PSL(2,\mathbb{R})$ as the matrices

$$\begin{array}{c}\hfill \pm \left(\begin{array}{cc}a& b\\ c& d\end{array}\right);a,b,c\mathrm{and}d\mathrm{are} \mathbb{R} \mathrm{and}ad-bc=1.\end{array}$$

As $PSL(2,\mathbb{R})$ acts on the upper half plane

$\mathbb{H}:=\left\{z\in \mathbb{C}:\mathrm{Im}\left(\mathrm{z}\right)>0\right\}$, $PSL(2,\mathbb{R})$ turns out to be the automorphism group of $\mathbb{H}$. We also denote the extended rationals $\mathbb{Q}\cup \{\infty \}$ by $\widehat{\mathbb{Q}}$ and $\mathbb{H}\cup \widehat{\mathbb{Q}}$ by $\mathcal{U}$.

The modular group $\mathsf{\Gamma }$ is the subgroup of $PSL(2,\mathbb{R})$ such that $a,b,c,$ and d are integers. The modular group has well-studied congruence subgroups. The principal congruence subgroup $\mathsf{\Gamma }\left(N\right)$, where N denotes a positive integer, of the modular group consists of the transformations corresponding to the matrices $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathsf{\Gamma }$ such that $a\equiv d\equiv 1modN$ and $b\equiv c\equiv 0modN$. The other congruence subgroups are ${\mathsf{\Gamma }}_1\left(N\right)$ and ${\mathsf{\Gamma }}_0\left(N\right)$. ${\mathsf{\Gamma }}_1\left(N\right)$ consists of the transformations corresponding to the matrices $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathsf{\Gamma }$ such that $a\equiv d\equiv 1modN$ and $c\equiv 0modN$, and ${\mathsf{\Gamma }}_0\left(N\right)$ consists of the transformations corresponding to the matrices $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathsf{\Gamma }$ such that $c\equiv 0modN$. The relations between these subgroups are well-known. $\mathsf{\Gamma }$ is the normalizer of $\mathsf{\Gamma }\left(N\right)$ in $PSL(2,\mathbb{R})$ and the normalizer of ${\mathsf{\Gamma }}_1\left(N\right)$ in $\mathsf{\Gamma }$ is ${\mathsf{\Gamma }}_0\left(N\right)$. The indices of these groups in relation to each other are given by the formulas

i.

$\mid \mathsf{\Gamma }:\mathsf{\Gamma }\left(N\right)\mid ={\displaystyle {\displaystyle \frac{N^3}{3}}\prod _{p\mid N}1-{\displaystyle \frac{1}{p^2}}}$,

ii.

$\mid {\mathsf{\Gamma }}_1\left(N\right):\mathsf{\Gamma }\left(N\right)\mid =N$,

iii.

$\mid {\mathsf{\Gamma }}_0\left(N\right):{\mathsf{\Gamma }}_1\left(N\right)\mid ={\displaystyle {\displaystyle \frac{N}{2}}\prod _{p\mid N}1-{\displaystyle \frac{1}{p}}}$,

and

i.

$\mid \mathsf{\Gamma }:\mathsf{\Gamma }\left(2\right)\mid =2$,

ii.

$\mid {\mathsf{\Gamma }}_1\left(2\right):\mathsf{\Gamma }\left(2\right)\mid =2$,

iii.

$\mid {\mathsf{\Gamma }}_0\left(N\right):{\mathsf{\Gamma }}_1\left(N\right)\mid =1$.

The study by Jones, Singerman and Wicks [1] is a pioneering study concerning these groups and has enabled the analytical examination of graphs. Many researchers have conducted studies [2,3,4,5,6,7] that reveal the relationship of many groups of graphswith the methods and results presented in this study. In particular, because of the interesting nature of the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ of ${\mathsf{\Gamma }}_0\left(N\right)$ in $PSL(2,\mathbb{R})$ [8,9,10] and its complexity relative to the modular group, researchers have studied normalizer-related graphs under various conditions [2,11,12,13].

Singerman, in his two studies [14,15], investigated the regular maps corresponding to the principal congruence subgroups of the modular group, arithmetically using the theory of maps [16] and universal tessellations [17]. The natural chain consisting of the modular group and its congruence subgroups allowed the author to construct regular maps, dividing the Farey map according to the principal congruence subgroup. Briefly, a map on an orientable surface is a decomposition of the surface into simply-connected polygonal cells called faces. Thus, a map is considered to have vertices and edges formed by the underlying graph and faces formed by the polygonal cells. It is known that a similar natural chain does not exist for the normalizer. However, the authors in [18] described two subgroups of the normalizer containing ${\mathsf{\Gamma }}_0\left(N\right)$ to obtain a chain, and investigated regular maps corresponding to the subgroups ${\mathsf{\Gamma }}_0\left(N\right)$ for the values of N, which make the normalizer a triangle group. Their first study [18] concerned the investigation of maps with triangular faces and their subsequent study [19] concerned maps with quadrilateral and hexagonal faces. Their results appear to be similar to those of the relation between $\mathsf{\Gamma }$ and $\mathsf{\Gamma }\left(N\right)$.

The main purpose of the present paper is to describe a modulo-N subgroup of the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ and to investigate regular maps corresponding to this modulo-N subgroup. In this manner we first construct a natural-like chain, and then construct regular maps, dividing the normalizer maps by the modulo-N subgroup ${\mathsf{\Gamma }}^h\left(N\right)$. This paper unifies maps with triangular, quadrilateral, and hexagonal faces. The results show that the regular maps that are constructed in this paper are quite interesting because, with the exception of some low values of N, they are all of large genus. When we deal with regular maps, we define three parametrizations, namely, $N_1,N_2,$ and $N_3$, each of which corresponds to a range of values that makes ${\mathsf{\Gamma }}_B\left(N\right)$ a triangular group. In this way we guarantee that all the maps are regular and we can reveal their arithmetic structure.

2. The Structure of the Normalizer and Some Subgroups

As described in [9], the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ of ${\mathsf{\Gamma }}_0\left(N\right)$ consists of the transformations corresponding to the matrices

$$\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)$$

(1)

where all symbols represent integers, h is the largest divisor of 24 for which $h^2\mid N$, $e>0$ is an exact divisor of $N/h^2$, and the determinant is e. (We say that r is an exact divisor of s if $r\mid s$ and $(r,s/r)=1$).

Here we define some subgroups of the normalizer, which helps us to form and investigate the maps.

The first group we are going to define is ${\mathsf{\Gamma }}^h\left(N\right)$. It consists of the transformations corresponding to the matrices

$$\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right),a\equiv d\equiv 1modN\mathit{and}b\equiv c\equiv 0mod\left(N\right),$$

where h is the largest divisor of 24 for which $h^2\mid N$.

Here we present a special subgroup of $\mathsf{\Gamma }\left(N\right)$ in order to calculate the index of ${\mathsf{\Gamma }}^h\left(N\right)$ in ${\mathsf{\Gamma }}_B\left(N\right)$. We denote this subgroup of $\mathsf{\Gamma }\left(N\right)$ by ${\mathsf{\Gamma }}^{\ast }\left(N\right)$, which consists of the transformations corresponding to the matrices $\left(\begin{array}{cc}a& bN\\ cN& d\end{array}\right)\in \mathsf{\Gamma }\left(N\right)$ such that $c\equiv 0modN/h^2$.

Proposition 1.

$\mid \mathsf{\Gamma }\left(N\right):{\mathsf{\Gamma }}^{\ast }\left(N\right)\mid =N/h^2$

Proof. 

The proof follows from the homomorphism $\phi :\mathsf{\Gamma }\left(N\right)\to {\mathbb{Z}}_{N/h^2}$ defined by $\phi \left(\begin{array}{cc}a& bN\\ cN& d\end{array}\right)=cmodN/h^2$ and the first isomorphism Theorem. □

Remark 1.

${\mathsf{\Gamma }}^h\left(N\right)$ is a conjugation of ${\mathsf{\Gamma }}^{\ast }\left(N\right)$ by $\left(\begin{array}{cc}h& 0\\ 0& 1\end{array}\right)$. One can easily verify this conjugation via the following equality

$${\mathsf{\Gamma }}^{\ast }\left(N\right)=\left(\begin{array}{cc}h& 0\\ 0& 1\end{array}\right){\mathsf{\Gamma }}^h\left(N\right)\left(\begin{array}{cc}1& 0\\ 0& h\end{array}\right).$$

(2)

The following remark directly follows from Remark 1.

Remark 2.

$\mid \mathsf{\Gamma }\left(N\right):{\mathsf{\Gamma }}^h\left(N\right)\mid =N/h^2$

The following proposition is one of the most important propositions in the paper because it will ensure that the maps constructed in the paper are all regular.

Proposition 2.

${\mathsf{\Gamma }}^h\left(N\right)$ is a normal subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$.

Proof. 

Let $T=\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)\in {\mathsf{\Gamma }}_B\left(N\right)$ and $S=\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\in {\mathsf{\Gamma }}^h\left(N\right)$. Consider

$$\begin{array}{cc}\hfill & T^{-1}ST=\left(\begin{array}{cc}de& -b/h\\ -cN/h& ae\end{array}\right)\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}adex-abzN/h^2+cdyN/h^2-bctN/e{h}^2& (bdx-b^{2}zN/e{h}^2+d^{2}ey-bdt)/h\\ (a^{2}ez-acx+act-c^{2}yN/e{h}^2)N/h& abzN/h^2-bcxN/e{h}^2+adet-cdyN/h^2\end{array}\right),\hfill \end{array}$$

where the resulting matrix is divided by e in order to obtain a matrix with determinant 1. Since $x\equiv t\equiv 1modN$, $y\equiv z\equiv 0modN$, using the determinant $ade-bcN/e{h}^2=1$ of T, we have

$$\begin{array}{cc}\hfill adex-abzN/h^2+cdyN/h^2-bctN/e{h}^2& \equiv 1modN\hfill \\ \hfill abzN/h^2-bcxN/e{h}^2+adet-cdyN& \equiv 1modN\hfill \\ \hfill bdx-b^{2}zN/e{h}^2+d^{2}ey-bdt& \equiv 0modN\hfill \\ \hfill a^{2}ez-acx+act-c^{2}yN/e{h}^2& \equiv 0modN.\hfill \end{array}$$

This completes the proof. □

Theorem 1

([20]). $\mid {\mathsf{\Gamma }}_B\left(N\right):{\mathsf{\Gamma }}_0\left(N\right)\mid =2^{\rho }{h}^2\tau$, where ρ is the number of distinct prime factors of $N/h^2$ and

$$\tau ={\displaystyle \frac{{\displaystyle \prod _{p\mid N}\left(1+{\displaystyle \frac{1}{p}}\right)}}{{\displaystyle \prod _{p\mid N/h^2}\left(1+{\displaystyle \frac{1}{p}}\right)}}}.$$

Finally, using Theorem 1 and Remark 2, we can present the following Proposition.

Proposition 3.

For $N>2$, $\mid {\mathsf{\Gamma }}_B\left(N\right):{\mathsf{\Gamma }}^h\left(N\right)\mid =2^{\rho -1}\eta N^3{\displaystyle \prod _{p\mid N}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$, and for $N=2$, $\mid {\mathsf{\Gamma }}_B\left(2\right):{\mathsf{\Gamma }}^1\left(2\right)\mid =8$, where ρ is the number of distinct prime factors of $N/h^2$ and

$$\eta ={\displaystyle \frac{1}{{\displaystyle \prod _{p\mid N/h^2}\left(1+{\displaystyle \frac{1}{p}}\right)}}}.$$

Proof. 

This is straightforward due to the relation

$$\mid {\mathsf{\Gamma }}_B\left(N\right):{\mathsf{\Gamma }}^h\left(N\right)\mid =\mid {\mathsf{\Gamma }}_B\left(N\right):{\mathsf{\Gamma }}_0\left(N\right)\mid \mid {\mathsf{\Gamma }}_0\left(N\right):\mathsf{\Gamma }\left(N\right)\mid \mid \mathsf{\Gamma }\left(N\right):{\mathsf{\Gamma }}^h\left(N\right)\mid .$$

If $N=2$, then since $\mid {\mathsf{\Gamma }}_0\left(2\right):\mathsf{\Gamma }\left(2\right)\mid =2$, we have $\mid {\mathsf{\Gamma }}_B\left(2\right):{\mathsf{\Gamma }}^1\left(2\right)\mid =8$. □

The second important subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$ that we described is ${\mathsf{\Gamma }}_1^h\left(N\right)$, which consists of the transformations corresponding to the matrices

$$\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right),a\equiv d\equiv 1modN\mathit{and}c\equiv 0modN,$$

where h is the largest divisor of 24 for which $h^2\mid N$.

Proposition 4.

${\mathsf{\Gamma }}^h\left(N\right)$ is a normal subgroup of ${\mathsf{\Gamma }}_1^h\left(N\right)$.

Proof. 

By definition, it is easily seen that ${\mathsf{\Gamma }}^h\left(N\right)$ is the kernel of the homomorphism $\psi :{\mathsf{\Gamma }}_1^h\left(N\right)\to {\mathbb{Z}}_N$ defined by

$$\psi \left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right)=bmodN.$$

□

Proposition 5.

$\mid {\mathsf{\Gamma }}_1^h\left(N\right):{\mathsf{\Gamma }}^h\left(N\right)\mid =N$

Proof. 

This is straightforward due to the homomorphism in the proof of Proposition 4 and the first isomorphism theorem. □

By means of Proposition 3 and Proposition 5, the following corollary is obtained.

Corollary 1.

For $N>2$, $\mid {\mathsf{\Gamma }}_B\left(N\right):{\mathsf{\Gamma }}_1^h\left(N\right)\mid =2^{\rho -1}\eta N^2{\displaystyle \prod _{p\mid N}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$, where ρ is the number of distinct prime factors of $N/h^2$ and

$$\eta ={\displaystyle \frac{1}{{\displaystyle \prod _{p\mid N/h^2}\left(1+{\displaystyle \frac{1}{p}}\right)}}}.$$

For $N=2$, $\mid {\mathsf{\Gamma }}_B\left(2\right):{\mathsf{\Gamma }}_1^1\left(2\right)\mid =4$.

The last subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$ that we described is ${\mathsf{\Gamma }}_B^h\left(N\right)$, which consists of the transformations corresponding to the matrices

$$\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right),c\equiv 0mode{h}^2,$$

where h is the largest divisor of 24 for which $h^2\mid N$ and the determinant of the matrix is e.

The purpose of defining this group is to be able to form a chain. In fact, this group corresponds to the axes of the maps, but axes will not be examined in this article since finding the axes is not requisite for revealing the structure of the map. So, just to complete the chain, we will give only the following theorem.

Proposition 6.

The normalizer of ${\mathsf{\Gamma }}_1^h\left(N\right)$ in ${\mathsf{\Gamma }}_B\left(N\right)$ is ${\mathsf{\Gamma }}_B^h\left(N\right)$.

Proof. 

Let $N\left({\mathsf{\Gamma }}_1^h\left(N\right)\right)$ denote the normalizer of ${\mathsf{\Gamma }}_1^h\left(N\right)$ in ${\mathsf{\Gamma }}_B\left(N\right)$. Consider an element $T=\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)$ of ${\mathsf{\Gamma }}_B^h\left(N\right)$; then, for $U=\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\in {\mathsf{\Gamma }}_1^h\left(N\right)$, we have

$$\begin{array}{cc}\hfill & T^{-1}UT=\left(\begin{array}{cc}de& -b/h\\ -cN/h& ae\end{array}\right)\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}ad{e}^{2}x-abezN/h^2+dceyN/h^2-bctN/h^2& {\displaystyle \frac{bdex-bdet+d^{2}y{e}^2-b^{2}zN/h^2}{h}}\\ (acet-acex+a^2{e}^{2}z-c^{2}yN/h^2)N/h& ad{e}^{2}t-bcxN/h^2+abzeN/h^2-cdeyN/h^2\end{array}\right).\hfill \end{array}$$

The resulting matrix has determinant $e^2$. Thus, dividing all terms by e, we have a matrix with determinant 1

$$\begin{array}{c}\hfill \left(\begin{array}{cc}adex-abzN/h^2+dcyN/h^2-bctN/e{h}^2& {\displaystyle \frac{bdx-bdt+d^{2}ye-b^{2}zN/e{h}^2}{h}}\\ (act-acx+a^{2}ez-c^{2}yN/e{h}^2)N/h& adet-bcxN/e{h}^2+abzN/h^2-cdyN/h^2\end{array}\right).\end{array}$$

Since $x\equiv t\equiv 1modN$, $c\equiv 0mode{h}^2$, $z\equiv 0modN$ and $ade-bcN/e{h}^2=1$, we have

$$adex-abzN/h^2+dcyN/h^2-bctN/e{h}^2\equiv adet-bcxN/e{h}^2+abzN/h^2-cdyN/h^2\equiv 1modN.$$

(3)

Furthermore, $c\equiv 0mode{h}^2$ and $z\equiv 0modN$ yield

$$act-acx+a^{2}ez-c^{2}yN/e{h}^2\equiv 0modN.$$

(4)

Thus, using (3) and (4), we conclude $TU{T}^{-1}\in {\mathsf{\Gamma }}_1^h\left(N\right)$, that is, $T\in N\left({\mathsf{\Gamma }}_1^h\left(N\right)\right)$.

Conversely, let $S=\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)\in N\left({\mathsf{\Gamma }}_1^h\left(N\right)\right)$. Then, for an arbitrary element $U=\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\in {\mathsf{\Gamma }}_1^h\left(N\right)$, we have $SU{S}^{-1}\in {\mathsf{\Gamma }}_1^h\left(N\right)$. Therefore,

$$\begin{array}{cc}\hfill & SU{S}^{-1}=\left(\begin{array}{cc}de& -b/h\\ -cN/h& ae\end{array}\right)\left(\begin{array}{cc}x& y/h\\ zN/h& t\end{array}\right)\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}ad{e}^{2}x-abezN/h^2+dceyN/h^2-bctN/h^2& {\displaystyle \frac{bdex-bdet+d^{2}y{e}^2-b^{2}zN/h^2}{h}}\\ (acet-acex+a^2{e}^{2}z-c^{2}yN/h^2)N/h& ad{e}^{2}t-bcxN/h^2+abzeN/h^2-cdeyN/h^2\end{array}\right),\hfill \end{array}$$

where the resulting matrix has determinant $e^2$. Dividing each element by e, we have

$$\left(\begin{array}{cc}adex-abzN/h^2+dcyN/h^2-bctN/e{h}^2& {\displaystyle \frac{bdx-bdt+d^{2}ye-b^{2}zN/e{h}^2}{h}}\\ (act-acx+a^{2}ez-c^{2}yN/e{h}^2)N/h& adet-bcxN/e{h}^2+abzN/h^2-cdyN/h^2\end{array}\right).$$

Since the resulting matrix is in ${\mathsf{\Gamma }}_1^h\left(N\right)$, we have

$$\begin{array}{cc}\hfill & adex-abzN/h^2+dcyN/h^2-bctN/e{h}^2\equiv 1modN\hfill \\ \hfill & adet-bcxN/e{h}^2+abzN/h^2-cdyN/h^2\equiv 1modN,\hfill \\ \hfill & act-acx+a^{2}ez-c^{2}yN/e{h}^2\equiv 0modN.\hfill \end{array}$$

Consider the congruences $adex-abzN/h^2+dcyN/h^2-bctN/e{h}^2\equiv 1modN$ and $adet-bcxN/e{h}^2+abzN/h^2-cdyN/h^2\equiv 1modN$. Using $ade-bcN/e{h}^2=1$, $x\equiv t\equiv 1modN$ and $z\equiv 0modN$, we obtain $cd\equiv 0mod{h}^2$. Moreover, consider the congruence $act-acx+a^{2}ez-c^{2}yN/e{h}^2\equiv 0modN$. Since $x\equiv t\equiv 1modN$, we have $x-t\equiv 0modN$. Thus, $z\equiv 0modN$ yields $c^{2}N/e{h}^2\equiv 0modN$. As $h^2\mid N$, we have $h^2\mid c^{2}N/e{h}^2$. Since $h^2\mid cd$ and $h^2\mid c^{2}N/e{h}^2$, for integers $de$ and b we have $h^2\mid adec-b{c}^{2}N/e{h}^2=c$. Finally, by $h^2\mid c$ and $N\mid c^{2}N/e{h}^2$, one can easily obtain $e{h}^2\mid c$. This means that $S\in {\mathsf{\Gamma }}_B^h\left(N\right)$. □

We now present two important theorems of [20] regarding the structure of ${\mathsf{\Gamma }}_B\left(N\right)$.

Theorem 2.

Let $N=2^{\alpha }{3}^{\beta }$ and $\beta =0$ or 2. Then, ${\mathsf{\Gamma }}_B\left(N\right)$ is a triangle group if and only if $\alpha \le 8$. In these cases

$${\mathsf{\Gamma }}_B\left(N\right)has signature\left\{\begin{array}{cc}(0;2,3,\infty )\hfill & \mathit{if}\alpha =0,2,4,6\hfill \\ (0;2,4,\infty )\hfill & \mathit{if}\alpha =1,3,5,7\hfill \\ (0;2,\infty ,\infty )\hfill & \mathit{if}\alpha =8.\hfill \end{array}\right.$$

Theorem 3.

Let $N=2^{\alpha }{3}^{\beta }$ and $\beta =1$ or 3. Then ${\mathsf{\Gamma }}_B\left(N\right)$ is a triangle group if and only if $\alpha =0,2,4,6$. In these cases ${\mathsf{\Gamma }}_B\left(N\right)$ has signature $(0;2,6,\infty )$.

Based on the above mentioned theorems we conclude

Remark 3.

Let $N=2^{\alpha }{3}^{\beta }$,

i. 

if $\beta =0,2$ and $\alpha =0,2,4,6$, then regular 3-valent maps correspond to normal subgroups of ${\mathsf{\Gamma }}_B\left(N\right)$,

ii. 

if $\beta =0,2$ and $\alpha =1,3,5,7$, then regular 4-valent maps correspond to normal subgroups of ${\mathsf{\Gamma }}_B\left(N\right)$,

iii. 

if $\beta =1,3$ and $\alpha =0,2,4,6$, then regular 6-valent maps correspond to normal subgroups of ${\mathsf{\Gamma }}_B\left(N\right)$.

In the present study we investigate the regular 3-valent, 4-valent, and 6-valent maps corresponding to normal subgroups of ${\mathsf{\Gamma }}_B\left(N\right)$.

Let us identify parameter N corresponding to each statement of Remark 3.

Notation 1.

We denote N by

i. 

$N_1$ if N satisfies i of Remark 3,

ii. 

$N_2$ if N satisfy $ii$ of Remark 3,

iii. 

$N_3$ if N satisfy $iii$ of Remark 3.

Let us identify the elements of ${\mathsf{\Gamma }}_B\left(N\right)$ for each parameter in Notation 1 in the following Remark.

Remark 4.

i. 

According to the definition of $N_1$, we have $N_1=h^2$ and hence $e=1$. Thus, ${\mathsf{\Gamma }}_B\left(N_1\right)$ consists of the transformations corresponding to the matrices

$$\left(\begin{array}{cc}a& b/h\\ ch& d\end{array}\right),ad-bc=1,$$

(5)

where h is the largest divisor of 24 for which $h^2\mid N_1$.

ii. 

According to the definition of $N_2$, we have $N_2=2h^2$ and hence $e=1,2$. Thus, ${\mathsf{\Gamma }}_B\left(N_2\right)$ consists of two types of elements, namely, even elements and odd elements [19]. Even elements are the transformations corresponding to the matrices

$$\left(\begin{array}{cc}a& b/h\\ 2ch& d\end{array}\right),ad-2bc=1,$$

(6)

and odd elements are the transformations corresponding to the matrices

$$\left(\begin{array}{cc}2a& b/h\\ 2ch& 2d\end{array}\right),2ad-bc=1,$$

(7)

where h is the largest divisor of 24 for which $h^2\mid N_2$.

iii. 

According to definition of $N_3$, we have $N_2=3h^2$ and hence $e=1,3$. Similarly to $ii$, ${\mathsf{\Gamma }}_B\left(N_3\right)$ consists of even elements and odd elements [19]. Even elements are the transformations corresponding to the matrices

$$\left(\begin{array}{cc}a& b/h\\ 3ch& d\end{array}\right),ad-3bc=1,$$

(8)

and odd elements are the transformations corresponding to the matrices

$$\left(\begin{array}{cc}3a& b/h\\ 3ch& 3d\end{array}\right),3ad-bc=1,$$

(9)

where h is the largest divisor of 24 for which $h^2\mid N_3$.

3. The Normalizer Maps

In this section, we construct three universal maps, namely, the normalizer maps ${\mathcal{M}}_3^h$, ${\mathcal{M}}_4^h$, and ${\mathcal{M}}_6^h$. These universal maps are investigated in [18,19]. On the other hand, Akbaş showed that ${\mathsf{\Gamma }}_B\left(N\right)$ acts transitively on $\widehat{\mathbb{Q}}$ for $N_1$, $N_2$, and $N_3$. Now we are ready to construct the normalizer maps.

First let us introduce the universal map ${\mathcal{M}}_3^h$ corresponding to ${\mathsf{\Gamma }}_B\left(N_1\right)$ (see Notation 1 for the parameters $N_1,N_2,$ and $N_3$). Vertices of ${\mathcal{M}}_3^h$ are the fractions ${\displaystyle \frac{a}{ch}}$ with $(a,c)=1$ and two vertices, ${\displaystyle \frac{a}{ch}}$ and ${\displaystyle \frac{b}{dh}}$, are joined by an edge if and only if $ad-bc=\pm 1$. All these edges are just hyperbolic lines, and in this way ${\mathcal{M}}_3^h$ has the following properties:

• There is a triangle with vertices ${\displaystyle \frac{1}{0}},{\displaystyle \frac{1}{h}},{\displaystyle \frac{0}{h}}$.
• ${\mathsf{\Gamma }}_B\left(N_1\right)$ acts as a group of homomorphisms of ${\mathcal{M}}_3^h$.
• There is a triangle with vertices ${\displaystyle \frac{a}{ch}},{\displaystyle \frac{a+b}{(c+d)h}},{\displaystyle \frac{b}{dh}}$.

${\mathcal{M}}_3^h$ is a triangular tessellation of the upper half plane. The triangles in 3 are the images of the triangle in 1 under the elements of ${\mathsf{\Gamma }}_B\left(N\right)$.

Remark 5.

One can easily see that when $h=1$, ${\mathcal{M}}_3^1$ is just the Farey map. When $h>1$, the normalizer map ${\mathcal{M}}_3^h$ is the Farey map scaled by a factor of $1/h$ (see Figure 1 for $N_1=4$ and $h=2$).

In a similar way, we can construct the normalizer map ${\mathcal{M}}_4^h$ (${\mathcal{M}}_6^h$) corresponding to ${\mathsf{\Gamma }}_B\left(N_2\right)$ (${\mathsf{\Gamma }}_B\left(N_3\right)$). However, there is a slight difference: since ${\mathsf{\Gamma }}_B\left(N_2\right)$ (${\mathsf{\Gamma }}_B\left(N_3\right)$) has two types of elements, namely, the even and odd elements, vertices of ${\mathcal{M}}_4^h$ (${\mathcal{M}}_6^h$) have two categories, namely, even and odd vertices. Even vertices are the fractions ${\displaystyle \frac{a}{2ch}}$ (${\displaystyle \frac{a}{3ch}}$) with $(a,c)=1$ and $2\nmid a$ ($(a,c)=1$ and $3\nmid a$). Odd vertices are the fractions ${\displaystyle \frac{b}{dh}}$ (${\displaystyle \frac{b}{dh}}$) with $(b,d)=1$ and $2\nmid d$ ($(b,d)=1$ and $3\nmid d$).

We can denote 0 and ∞ by ${\displaystyle \frac{0}{1.h}}$ (resp. ${\displaystyle \frac{0}{1.h}}$) and ${\displaystyle \frac{1}{2.0.h}}$ (resp. ${\displaystyle \frac{1}{3.0.h}}$), respectively. In this manner, we can see that ∞ is an even vertex and 0 is an odd vertex. Thus the edges of ${\mathcal{M}}_4^h$ (resp. ${\mathcal{M}}_6^h$) are the images of the edge joining 0 and ∞ under the elements of ${\mathsf{\Gamma }}_B\left(N_2\right)$ (resp. ${\mathsf{\Gamma }}_B\left(N_3\right)$). Hence all the edges of ${\mathcal{M}}_4^h$ (resp. ${\mathcal{M}}_6^h$) join an even and odd vertex. Finally, the vertices ${\displaystyle \frac{a}{2ch}}$ (resp. ${\displaystyle \frac{a}{3ch}}$) and ${\displaystyle \frac{b}{dh}}$ (resp. ${\displaystyle \frac{b}{dh}}$) are joined by an edge if and only if $ad-2bc=\pm 1$ (resp. $ad-3bc=\pm 1$). We know that ${\mathcal{M}}_4^h$ (resp. ${\mathcal{M}}_6^h$) is quadrilateral (resp. hexagonal), and its principal quadrilateral (resp. hexagon) is $\infty ,0,{\displaystyle \frac{1}{2h}},{\displaystyle \frac{1}{h}}$ (resp. $\infty ,0,{\displaystyle \frac{1}{3h}},{\displaystyle \frac{1}{2h}},{\displaystyle \frac{2}{3h}},{\displaystyle \frac{1}{h}}$). One can find other quadrilaterals (resp. hexagons) applying the elements of ${\mathsf{\Gamma }}_B\left(N_2\right)$ (resp. ${\mathsf{\Gamma }}_B\left(N_3\right)$) to the vertices of the principle quadrilateral (resp. hexagon).

In [17], the author showed that any triangular map on a surface is the quotient of the universal triangular map by a subgroup of $\mathsf{\Gamma }(2,\infty ,3)$ and any regular triangular map is the quotient of the universal triangular map by a normal subgroup of $\mathsf{\Gamma }(2,\infty ,3)$. As ${\mathsf{\Gamma }}^h\left(N\right)$ is a normal subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$ and ${\mathsf{\Gamma }}_B\left(N\right)$ is a triangle group for all values of $N_1$, $N_2$, and $N_3$, we can form the regular maps ${\mathcal{M}}_3^h/{\mathsf{\Gamma }}^h\left(N_1\right)$, ${\mathcal{M}}_4^h/{\mathsf{\Gamma }}^h\left(N_2\right)$, and ${\mathcal{M}}_6^h/{\mathsf{\Gamma }}^h\left(N_3\right)$. Since ${\mathsf{\Gamma }}^h\left(N\right)$ is a modulo-N-subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$, we will call these regular maps “normalizer maps modulo N”.

4. Normalizer Maps Modulo N

Definition 1.

The map ${\mathcal{M}}_i^h\left(N\right)$ is defined as the map ${\mathcal{M}}_i^h/{\mathsf{\Gamma }}^h\left(N\right)$ for $i=3,4,6$.

Before investigating regular maps ${\mathcal{M}}_i^h\left(N\right)$, $i=3,4,6$ analytically, we present some results concerning the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ and its subgroups that we already described in Section 2.

A straightforward calculation yields that the stabilizer $S_{\infty }$ of ∞ in ${\mathsf{\Gamma }}_B\left(N\right)$ is the cyclic group generated by $\left(\begin{array}{cc}1& 1/h\\ 0& 1\end{array}\right)$. Let $[\infty ]$ be the orbit of ∞ under the action of ${\mathsf{\Gamma }}^h\left(N\right)$ and $S_{[\infty ]}$ be the set-wise stabilizer of $[\infty ]$ in ${\mathsf{\Gamma }}_B\left(N\right)$.

Proposition 7.

$S_{[\infty ]}=S_{\infty }{\mathsf{\Gamma }}^h\left(N\right)$.

Proof. 

Let $T\in S_{[\infty ]}$. For an arbitrary element U of ${\mathsf{\Gamma }}^h\left(N\right)$, since $U\infty \in [\infty ]$, T stabilizes $U\infty$. So we have $TU\infty \in [\infty ]$. That is, there exists an element $V\in {\mathsf{\Gamma }}^h\left(N\right)$ such that $TU\infty =V\infty$. This means that $V^{-1}TU\in S_{\infty }$. As ${\mathsf{\Gamma }}^h\left(N\right)$ is a normal subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$; we thus have $UT=TU$. Then $R^{-1}T\in S_{\infty }$, where $R=U^{-1}V\in {\mathsf{\Gamma }}^h\left(N\right)$. Finally, we conclude that $T\in {\mathsf{\Gamma }}^h\left(N\right)S_{\infty }=S_{\infty }{\mathsf{\Gamma }}^h\left(N\right)$, as ${\mathsf{\Gamma }}^h\left(N\right)$ is normal in ${\mathsf{\Gamma }}_B\left(N\right)$.

For the converse, let $T\in S_{\infty }{\mathsf{\Gamma }}^h\left(N\right)$. Then, there exist $U\in S_{\infty }$ and $V\in {\mathsf{\Gamma }}^h\left(N\right)$ such that $T=UV$. Consider the action of $UV$ on $[\infty ]$. As $V\in {\mathsf{\Gamma }}^h\left(N\right)$ and U stabilizes ∞, we have $T[\infty ]=UV[\infty ]=[\infty ]$. □

Proposition 8.

$S_{\infty }{\mathsf{\Gamma }}^h\left(N\right)={\mathsf{\Gamma }}_1^h\left(N\right)$.

Proof. 

Let $T\in S_{\infty }{\mathsf{\Gamma }}^h\left(N\right)$. Then, there exist $U=\left(\begin{array}{cc}1& u/h\\ 0& 1\end{array}\right)\in S\infty$ and $V=\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right)\in {\mathsf{\Gamma }}^h\left(N\right)$ such that $T=UV$. Consider the following matrix

$$T=UV=\left(\begin{array}{cc}1& u/h\\ 0& 1\end{array}\right)\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right)=\left(\begin{array}{cc}a+cuN/h^2& (b+du)/h\\ cN/h& d\end{array}\right).$$

As $a\equiv 1modN$ and $c\equiv 0modN$, we have $a+cuN/h^2\equiv 1modN$. Furthermore, $d\equiv 1modN$ yields that the resulting matrix is in ${\mathsf{\Gamma }}_1^h\left(N\right)$.

For the converse, let $T=\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right)\in {\mathsf{\Gamma }}_1^h\left(N\right)$. Consider the following equality

$$\left(\begin{array}{cc}1& ab/h\\ 0& 1\end{array}\right)\left(\begin{array}{cc}a-abcN/h^2& b(1-ad)/h\\ cN/h& d\end{array}\right)=\left(\begin{array}{cc}a& b/h\\ cN/h& d\end{array}\right)=T.$$

Since $a\equiv 1modN$ and $c\equiv 0modN$, $a-abcN/h^2\equiv 1modN$. Furthermore, $a\equiv d\equiv 1modN$ yields $b(1-ad)\equiv 0modN$. Thus,

$$\left(\begin{array}{cc}a-abcN/h^2& b(1-ad)/h\\ cN/h& d\end{array}\right)\in {\mathsf{\Gamma }}_1^h\left(N\right).$$

This completes the proof. □

Now we are ready to find the number of the vertices of ${\mathcal{M}}_i^h\left(N\right)$, $i=3,4,6$. We give the theorem only for ${\mathcal{M}}_3^h\left(N_1\right)$. This theorem also holds for ${\mathcal{M}}_4^h\left(N_2\right)$ and ${\mathcal{M}}_6^h\left(N_2\right)$.

Theorem 4.

There exists a one-to-one correspondence between the left cosets of ${\mathsf{\Gamma }}_1^h\left(N_1\right)$ in ${\mathsf{\Gamma }}_B\left(N_1\right)$ and the vertices of ${\mathcal{M}}_3^h\left(N_1\right)$.

Proof. 

Due to the transitivity of the action of ${\mathsf{\Gamma }}_B\left(N_1\right)$ on $\widehat{\mathbb{Q}}$ and ${\mathsf{\Gamma }}_1^h\left(N_1\right)⊴{\mathsf{\Gamma }}_B\left(N_1\right)$, we can choose the fixed vertex $[\infty ]$ of ${\mathcal{M}}_3^h\left(N_1\right)$. In accordance with Propositions 7 and 8, stabilizer of the vertex $[\infty ]$ in ${\mathsf{\Gamma }}_B\left(N_1\right)$ is ${\mathsf{\Gamma }}_1^h\left(N_1\right)$. Therefore, the theorem follows from the orbit-stabilizer theorem. □

Using Theorem 4 and Corollary 1, we can give the following corollary.

Corollary 2.

The number of vertices is

i. 

$V_3^h\left(N_1\right)=2^{\rho -1}\eta N_1^2{\displaystyle \prod _{p\mid N_1}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_3^h\left(N_1\right)$;

ii. 

$V_4^h\left(N_2\right)=2^{\rho -1}\eta N_2^2{\displaystyle \prod _{p\mid N_2}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_4^h\left(N_2\right)$;

iii. 

$V_6^h\left(N_3\right)=2^{\rho -1}\eta N_3^2{\displaystyle \prod _{p\mid N_3}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_6^h\left(N_3\right)$.

Now, we find that the set of left cosets has a one-to-one correspondence with the set of vertices of the regular maps. So if we want to study the vertices, we can study the left cosets. Therefore, the following propositions will allow us to identify the vertices uniquely.

Proposition 9.

Let $T_1=\left(\begin{array}{cc}{a}_1{e}_1& b_1/h\\ c_{1}N/h& d_1{e}_1\end{array}\right),T_2\left(\begin{array}{cc}{a}_2{e}_2& b_2/h\\ c_{2}N/h& d_2{e}_2\end{array}\right)\in {\mathsf{\Gamma }}_B\left(N\right)$. Then, $T_1$ and $T_2$ determine the same left coset of ${\mathsf{\Gamma }}_1^h\left(N\right)$ in ${\mathsf{\Gamma }}_B\left(N\right)$ if and only if $e_1=e_2$ and $\left(\begin{array}{c}{a}_{1}e\\ c_{1}N/e{h}^2\end{array}\right)\equiv \pm \left(\begin{array}{c}{a}_{2}e\\ c_{2}N/e{h}^2\end{array}\right)modN$.

Proof. 

If $e_1\ne e_2$ then the determinant of $T_2^{-1}{T}_1$ cannot be 1 such that $T_2^{-1}{T}_1\in {\mathsf{\Gamma }}_1^h\left(N\right)$. Thus, let $e_1=e_2=e$. Thus, $T_2^{-1}{T}_1\in {\mathsf{\Gamma }}_1^h\left(N\right)$ if and only if $\left(\begin{array}{c}{a}_1\\ c_1\end{array}\right)\equiv \pm \left(\begin{array}{c}{a}_2\\ c_2\end{array}\right)modN$ which follows from the definition of ${\mathsf{\Gamma }}_1^h\left(N\right)$. □

Proposition 9 states that each e determines different types of vertices. For the map ${\mathcal{M}}_3^h\left(N_1\right)$, since $e=1$, we have just one type of vertex. So we simply call them vertices. However, for ${\mathcal{M}}_4^h\left(N_2\right)$ (${\mathcal{M}}_6^h\left(N_3\right)$), since $e=1$ or $e=2$ ($e=1$ or $e=3$), there are two types of vertices. So we call them, as before, the even vertices and odd vertices of the map. Here, let us identify these vertices using Proposition 9.

We denote the left coset of ${\mathsf{\Gamma }}_1^h\left(N\right)$ corresponding to the matrix $\left(\begin{array}{cc}ae& b/h\\ cN/h& de\end{array}\right)$ by the row vector $(ae,cN/e{h}^2)$ for the corresponding e, as the equivalence of cosets depends only on the integers $a,c$, and e. Since the determinant of the matrix is written in the form of $ade-bcn/e{h}^2=1$, we have that $(ae,cN/e{h}^2)=1$. So we denote the vertices of the maps by $(ae,cN/e{h}^2)$, as any coset corresponds to a vertex of the map. In the case of $N_1,N_2,N_3$, i.e., the case of ${\mathcal{M}}_3^h\left(N_1\right)$, ${\mathcal{M}}_4^h\left(N_2\right)$, and ${\mathcal{M}}_6^h\left(N_3\right)$, using (5)–(9), we have the following cases.

• For the case $N_1$, since $N_1=h^2$, we have $e=1$. Thus, using Proposition 9, the set of vertices of ${\mathcal{M}}_3^h\left(N_1\right)$ is

  $$\left\{(a,c)\mid a,c\in {\mathbb{Z}}_{N_1},(a,c,N_1)=1\right\}/\sim ,$$

  where $(a,c)\sim (N_1-a,N_1-c)$.
• For the case $N_2$, since $N_2=2h^2$, we have $e=1$ or $e=2$. Thus, there are two types of vertices, namely, the odd vertices and even vertices. Using Proposition 9, the set of odd vertices of ${\mathcal{M}}_4^h\left(N_2\right)$ is

  $$\left\{(2a,c)\mid a,c\in {\mathbb{Z}}_{N_2},(a,c,N_2)=1,2\nmid c\right\}/\sim ,$$

  where $(2a,c)\sim (2(N_2-a),N_2-c)$.
  The set of even vertices of ${\mathcal{M}}_4^h\left(N_2\right)$ is

  $$\left\{(a,2c)\mid a,c\in {\mathbb{Z}}_{N_2},(a,c,N_2)=1,2\nmid a\right\}/\sim ,$$

  where $(a,2c)\sim (N_2-a,2(N_2-c))$.
• For the case $N_3$, since $N_3=3h^2$, we have $e=1$ or $e=3$. Thus, there are two types of vertices, namely, the odd vertices and even vertices. Using Proposition 9, the set of odd vertices of ${\mathcal{M}}_6^h\left(N_3\right)$ is

  $$\left\{(3a,c)\mid a,c\in {\mathbb{Z}}_{N_3},(a,c,N_3)=1,3\nmid c\right\}/\sim ,$$

  where $(3a,c)\sim (3(N_3-a),N_3-c)$.
  The set of even vertices of ${\mathcal{M}}_6^h\left(N_3\right)$ is

  $$\left\{(a,3c)\mid a,c\in {\mathbb{Z}}_{N_3},(a,c,N_3)=1,3\nmid a\right\}/\sim ,$$

  where $(a,3c)\sim (N_3-a,3(N_3-c))$.

A directed edge of a map will be called a dart. Since ${\mathsf{\Gamma }}_B\left(N\right)$ for all parameters $N_1,N_2$, and $N_3$ acts transitively on $\widehat{\mathbb{Q}}$, it acts transitively on the darts of ${\mathcal{M}}_i^h$ for $i=3,4,6$. Thus, ${\mathsf{\Gamma }}_B\left(N\right)/{\mathsf{\Gamma }}^h\left(N\right)$ acts transitively on the darts of ${\mathcal{M}}_i^h\left(N\right)$ for $i=3,4,6$ and $N_1,N_2,N_3$ because ${\mathsf{\Gamma }}^h\left(N\right)$ is a normal subgroup of ${\mathsf{\Gamma }}_B\left(N\right)$. According to the results of [16], a map is regular if its automorphism group acts transitively on its darts, which makes ${\mathcal{M}}_i^h\left(N\right)$ regular for $i=3,4,6$ and $N_1,N_2,N_3$.

Here we present a theorem to find the number of darts of regular maps. We give the theorem only for ${\mathcal{M}}_3^h\left(N_1\right)$. Again, the theorem also holds for ${\mathcal{M}}_4^h\left(N_2\right)$ and ${\mathcal{M}}_6^h\left(N_3\right)$.

Theorem 5.

There exists a one-to-one correspondence between the left cosets of ${\mathsf{\Gamma }}^h\left(N_1\right)$ in ${\mathsf{\Gamma }}_B\left(N_1\right)$ and the darts of ${\mathcal{M}}_3^h\left(N_1\right)$.

Proof. 

Due to the transitivity of the action of ${\mathsf{\Gamma }}_B\left(N_1\right)$ on $\widehat{\mathbb{Q}}$ and ${\mathsf{\Gamma }}_h\left(N_1\right)⊴{\mathsf{\Gamma }}_B\left(N_1\right)$, we can choose the fixed dart $\left[0\right]\to [\infty ]$ of ${\mathcal{M}}_3^h\left(N_1\right)$. We will show that the stabilizer of the dart $\left[0\right]\to [\infty ]$ is ${\mathsf{\Gamma }}^h\left(N_1\right)$. Let $T{\mathsf{\Gamma }}_B\left(N_1\right)$ stabilize the dart $\left[0\right]\to [\infty ]$. Thus, it stabilizes both $\left[0\right]$ and $[\infty ]$. If T stabilizes $[\infty ]$, by Propositions 7 and 8, $T\in {\mathsf{\Gamma }}_1^h\left(N_1\right)$. On the other hand, if T stabilizes $\left[0\right]$, then for an arbitrary $S\in {\mathsf{\Gamma }}^h\left(N_1\right)$ we have $TS\in {\mathsf{\Gamma }}^h\left(N_1\right)$. Let $T=\left(\begin{array}{cc}{a}_1& b_1/h\\ c_1{N}_1/h& d_1\end{array}\right){\mathsf{\Gamma }}_1^h\left(N_1\right)$ and $S=\left(\begin{array}{cc}{a}_2& b_2/h\\ c_2{N}_1/h& d_2\end{array}\right)\in {\mathsf{\Gamma }}^h\left(N_1\right)$. Consider the following equation

$$TS=\left(\begin{array}{cc}{a}_1& b_1/h\\ c_1{N}_1/h& d_1\end{array}\right)\left(\begin{array}{cc}{a}_2& b_2/h\\ c_2{N}_1/h& d_2\end{array}\right)=\left(\begin{array}{cc}{a}_1{a}_2+b_1{c}_2{N}_1/h^2& (a_1{b}_2+b_1{d}_2)/h\\ (a_2{c}_1+c_2{d}_1)N_1/h& c_1{b}_2{N}_1/h^2+d_1{d}_2\end{array}\right).$$

(10)

The matrix on the right-hand side of Equation (10) must be in ${\mathsf{\Gamma }}^h\left(N_1\right)$. According to the definition of ${\mathsf{\Gamma }}^h\left(N_1\right)$ and ${\mathsf{\Gamma }}_1^h\left(N_1\right)$ we have $a_1{a}_2+b_1{c}_2{N}_1/h^2\equiv c_1{b}_2{N}_1/h^2+d_1{d}_2\equiv 1mod{N}_1$, and $a_2{c}_1+c_2{d}_1\equiv 0mod{N}_1$. Thus, $a_1{b}_2+b_1{d}_2\equiv 0mod{N}_1$ must hold. Since $b_2\equiv 0mod{N}_1$ and S is arbitrary, we have $b_1\equiv 0mod{N}_1$. This yields $T\in {\mathsf{\Gamma }}^h\left(N_1\right)$. Again, the orbit-stabilizer theorem applies. □

Based on Theorem 5 and Proposition 3, the following corollary follows.

Corollary 3.

The number of darts is

i. 

$D_3^h\left(N_1\right)=2^{\rho -1}\eta N_1^3{\displaystyle \prod _{p\mid N_1}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_3^h\left(N_1\right)$;

ii. 

$D_4^h\left(N_2\right)=2^{\rho -1}\eta N_2^3{\displaystyle \prod _{p\mid N_2}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_4^h\left(N_2\right)$;

iii. 

$D_6^h\left(N_3\right)=2^{\rho -1}\eta N_3^3{\displaystyle \prod _{p\mid N_3}\left(1-{\displaystyle \frac{1}{p^2}}\right)}$ for ${\mathcal{M}}_6^h\left(N_3\right)$.

Using the regularity of the maps, the definition of a dart and Corollary 3, with the following corollary, we can determine the number of edges and the number of faces of the maps. The number of edges is always the half of the number of darts, and the number of faces can be found by dividing the number of darts by the number of edges in each face of the map.

Corollary 4.

The number of edges is

i. 

$E_3^h\left(N_1\right)={\displaystyle \frac{D_3^h\left(N_1\right)}{2}}$ for ${\mathcal{M}}_3^h\left(N_1\right)$;

ii. 

$E_4^h\left(N_2\right)={\displaystyle \frac{D_4^h\left(N_2\right)}{2}}$ for ${\mathcal{M}}_4^h\left(N_2\right)$;

iii. 

$E_6^h\left(N_3\right)={\displaystyle \frac{D_6^h\left(N_3\right)}{2}}$ for ${\mathcal{M}}_6^h\left(N_3\right)$.

Corollary 5.

The number of faces is

i. 

$F_3^h\left(N_1\right)={\displaystyle \frac{D_3^h\left(N_1\right)}{3}}$ for ${\mathcal{M}}_3^h\left(N_1\right)$;

ii. 

$F_4^h\left(N_2\right)={\displaystyle \frac{D_4^h\left(N_2\right)}{4}}$ for ${\mathcal{M}}_4^h\left(N_2\right)$;

iii. 

$F_6^h\left(N_3\right)={\displaystyle \frac{D_6^h\left(N_3\right)}{6}}$ for ${\mathcal{M}}_6^h\left(N_3\right)$.

Finally, using the Euler formula $V-E+F=2-2g$ for a regular map, where $V,E,F$ is the numbers of vertices, edges, and faces of the map, respectively, we can find the genus of the map as follows.

Corollary 6.

The genus is computed using the formula

i. 

$g_3^h\left(N_1\right)=N_1^2{\displaystyle \frac{N_1-6}{3}}{2}^{\rho -3}\eta {\displaystyle \prod _{p\mid N_1}\left(1-{\displaystyle \frac{1}{p^2}}\right)}+1$ for ${\mathcal{M}}_3^h\left(N_1\right)$;

ii. 

$g_4^h\left(N_2\right)=N_2^2(N_2-4)2^{\rho -4}\eta {\displaystyle \prod _{p\mid N_2}\left(1-{\displaystyle \frac{1}{p^2}}\right)}+1$, $N>2$ and $g_4^1\left(2\right)=0$, $N=2$ for ${\mathcal{M}}_4^h\left(N_2\right)$;

iii. 

$g_6^h\left(N_3\right)=N_3^2{\displaystyle \frac{N_3-3}{3}}{2}^{\rho -2}\eta {\displaystyle \prod _{p\mid N_3}\left(1-{\displaystyle \frac{1}{p^2}}\right)}+1$ for ${\mathcal{M}}_6^h\left(N_3\right)$.

In the following section, we present some examples to show the details.

Examples for Low N

Example 1

(Triangular case). For $N_1=4$, we have $h=2$ and the regular map ${\mathcal{M}}_3^2\left(4\right)$. The numbers of vertices, edges, and faces are $V_3^2\left(4\right)=6$, $E_3^2\left(4\right)=12$, and $F_3^2\left(4\right)=8$, respectively. Genus of ${\mathcal{M}}_3^2\left(4\right)$ is $g_3^2\left(4\right)=0$. The vertices are

$$\begin{array}{c}\hfill (1,0),(0,1),(1,1),(1,2),(2,1),(3,1).\end{array}$$

This is an octahedron lying on the sphere $\mathcal{U}/{\mathsf{\Gamma }}^2\left(4\right)$ with Schläfli $\{3,4\}$ (see Figure 2).

Example 2

(Quadrilateral case 1). For $N_2=2$, we have $h=1$ and the regular map ${\mathcal{M}}_4^1\left(2\right)$. The numbers of vertices, edges, and faces are $V_4^1\left(2\right)=4$, $E_4^1\left(2\right)=4$, and $F_4^1\left(2\right)=2$, respectively. The genus of ${\mathcal{M}}_4^1\left(2\right)$ is $g_4^1\left(2\right)=0$. The odd vertices are

$$\begin{array}{c}\hfill (0,1),(2,1),\end{array}$$

and the even vertices are

$$\begin{array}{c}\hfill (1,0),(1,2).\end{array}$$

This is a di-square lying on the sphere $\mathcal{U}/{\mathsf{\Gamma }}^1\left(2\right)$ with Schläfli $\{4,2\}$ (see Figure 3).

Example 3

(Quadrilateral case 2). For $N_2=8$, we have $h=2$ and the regular map ${\mathcal{M}}_4^2\left(8\right)$. The numbers of vertices, edges, and faces are $V_4^2\left(8\right)=32$, $E_4^2\left(8\right)=128$, and $F_4^2\left(8\right)=64$, respectively. The genus of ${\mathcal{M}}_4^2\left(8\right)$ is $g_4^2\left(8\right)=17$. The odd vertices are

$$\begin{array}{cc}\hfill & (0,1),(0,3),(2,1),(2,3),(2,5),(2,7),(4,1),(4,3),\hfill \\ \hfill & (4,5),(4,7),(6,1),(6,3),(6,5),(6,7),(8,1),(8,3),\hfill \end{array}$$

and the even vertices are

$$\begin{array}{cc}\hfill & (1,0),(1,2),(1,4),(1,6),(1,8),(1,10),(1,12),(1,14),\hfill \\ \hfill & (3,0),(3,2),(3,4),(3,6),(3,8),(3,10),(3,12),(3,14).\hfill \end{array}$$

This is the map $R17.6$ lying on $\mathcal{U}/{\mathsf{\Gamma }}^2\left(8\right)$ with Schlafli $\{4,8\}$.

Example 4

(Hexagonal case). For $N_3=3$, we have $h=1$ and the regular map ${\mathcal{M}}_6^1\left(3\right)$. The numbers of vertices, edges, and faces are $V_6^1\left(3\right)=6$, $E_6^1\left(3\right)=9$, and $F_6^1\left(3\right)=3$, respectively. The genus of ${\mathcal{M}}_6^1\left(3\right)$ is $g_6^1\left(3\right)=1$. The odd vertices are

$$\begin{array}{c}\hfill (0,1),(3,1),(3,2),\end{array}$$

and the even vertices are

$$\begin{array}{c}\hfill (1,0),(1,3),(1,6).\end{array}$$

This is the map ${\{6,3\}}_{(0,2)}$ lying on the torus $\mathcal{U}/{\mathsf{\Gamma }}^1\left(3\right)$ with Schlafli $\{6,3\}$ (see Figure 4).

5. Complete Tables of the Regular Maps

In this final section we provide the complete tables of regular maps corresponding to $N_1,N_2,N_3$. The triangular maps are provided in

Table 1. The complete table of regular maps for $N_1$.

${\mathit{N}}_1$	D	E	V	F	g	Map (Schläfli)
$2^2$	$2^3·3$	$2^2·3$	$2·3$	$2^3$	0	$\{3,4\}$
$3^2$	$2^2·3^4$	$2·3^4$	$2^2·3^2$	$2^2·3^3$	10	$\{3,9\}$
$2^4$	$2^9·3$	$2^8·3$	$2^5·3$	$2^9$	81	$\{3,16\}$
$2^2·3^2$	$2^6·3^5$	$2^5·3^5$	$2^4·3^3$	$2^6·3^4$	$2^9·3^8·5+1$	$\{3,36\}$
$2^6$	$2^{15}·3$	$2^{14}·3$	$2^9·3$	$2^{15}$	$2^8·29+1$	$\{3,64\}$
$2^4·3^2$	$2^{12}·3^5$	$2^{11}·3^5$	$2^8·3^3$	$2^{12}·3^4$	$2^7·3^3·23+1$	$\{3,144\}$
$2^6·3^2$	$2^{18}·3^5$	$2^{17}·3^5$	$2^{12}·3^3$	$2^{18}·3^4$	$2^{11}·3^3·5·19+1$	$\{3,576\}$

.

The quadrilateral maps are provided in

Table 2. The complete table of regular maps for $N_2$.

${\mathit{N}}_2$	D	E	V	F	g	Map (Schläfli)
2	$2^3$	$2^2$	$2^2$	2	0	$\{4,2\}$
$2^3$	$2^8$	$2^7$	$2^5$	$2^6$	17	$\{4,8\}$
$2·3^2$	$2^5·3^4$	$2^4·3^4$	$2^4·3^2$	$2^3·3^4$	$2^2·3^2·7+1$	$\{4,18\}$
$2^5$	$2^{14}$	$2^{13}$	$2^9$	$2^{12}$	$2^8·7+1$	$\{4,32\}$
$2^3·3^2$	$2^{11}·3^4$	$2^{10}·3^4$	$2^8·3^2$	$2^9·3^4$	$2^7·3^2·17+1$	$\{4,72\}$
$2^7$	$2^{20}$	$2^{19}$	$2^{13}$	$2^{18}$	$2^{12}·31+1$	$\{4,128\}$
$2^5·3^2$	$2^{17}·3^7$	$2^{16}·3^7$	$2^{12}·3^5$	$2^{15}·3^7$	$2^{11}·3^4·71+1$	$\{4,288\}$
$2^7·3^2$	$2^{23}·3^4$	$2^{22}·3^4$	$2^{16}·3^2$	$2^{21}·3^4$	$2^{15}·3^2·277+1$	$\{4,1152\}$

.

The hexagonal maps are provided in

Table 3. The complete table of regular maps for $N_3$.

${\mathit{N}}_3$	D	E	V	F	g	Map (Schläfli)
3	$2·3^2$	$3^2$	$2·3$	3	1	$\{6,3\}$
$2^2·3$	$2^5·3^3$	$2^4·3^3$	$2^3·3^2$	$2^4·3^2$	$2^2·3^3+1$	$\{6,12\}$
$3^3$	$2·3^8$	$3^8$	$2·3^5$	$3^7$	$3^5·2^3+1$	$\{6,27\}$
$2^4·3$	$2^{11}·3^3$	$2^{10}·3^3$	$2^7·3^2$	$2^{10}·3^2$	$2^6·3^3·5+1$	$\{6,48\}$
$2^2·3^3$	$2^5·3^9$	$2^4·3^9$	$2^3·3v$	$2^4·3^8$	$2^2·3^6·11·13+1$	$\{6,108\}$
$2^6·3$	$2^{17}·3^3$	$2^{16}·3^3$	$2^{11}·3^2$	$2^{16}·3^2$	$2^{10}·3^2·31+1$	$\{6,192\}$
$2^4·3^3$	$2^{11}·3^9$	$2^{10}·3^9$	$2^7·3^6$	$2^{10}·3^8$	$2^6·3^6·11·13+1$	$\{6,432\}$
$2^6·3^3$	$2^{17}·3^9$	$2^{16}·3^9$	$2^{11}·3^6$	$2^{16}·3^8$	$2^{17}·3^9·5^2·23+1$	$\{6,1728\}$

.

6. Conclusions

In the present paper, we construct a method that reveals the structure of the maps modulo N corresponding to a modulo N subgroup of the normalizer. It is known that the structure of the normalizer is complicated compared to known modular groups. For this reason, the structure of the normalizer has been an active area of study for researchers in this field. When it comes to investigating regular maps, it is again interesting to investigate the normalizer as a universal map because of its variability. For instance, if one uses the modular group $\mathsf{\Gamma }$ as a universal map to form regular maps, the regular maps will be all triangular, but the normalizer admits triangular, quadrilateral, and hexagonal maps corresponding to its normal subgroups. In this article, we investigated triangular, quadrilateral, and hexagonal maps corresponding to a modulo N subgroup thanks to this rich structure. The results demonstrate that all of the resulting regular maps are of large genus except for the lower values of N. We formulated all of the elements of the regular maps, such as the number of vertices, edges, and darts, and the genus.