Normalizer Maps Modulo N

: The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer Γ B ( N ) of Γ 0 ( N ) 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 .


Introduction
We will denote the group of all linear fractional transformations by PSL(2, R). By a linear fractional transformation we mean a function of a complex variable z, defined by where a, b, c and d are R and ad − bc = 1.
It is well-known that the linear fractional transformations can be represented by matrices. Hence, we can regard the elements of PSL(2, R) as the matrices ± a b c d ; a, b, c and d are R and ad − bc = 1.
As PSL(2, R) acts on the upper half plane H := {z ∈ C : Im(z) > 0}, PSL(2, R) turns out to be the automorphism group of H. We also denote the extended rationals Q ∪ {∞} byQ and H ∪Q by U .
The modular group Γ is the subgroup of PSL(2, R) such that a, b, c, and d are integers. The modular group has well-studied congruence subgroups. The principal congruence subgroup Γ(N), where N denotes a positive integer, of the modular group consists of the transformations corresponding to the matrices a b c d ∈ Γ such that a ≡ d ≡ 1 mod N and b ≡ c ≡ 0 mod N. The other congruence subgroups are Γ 1 (N) and Γ 0 (N). 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 Γ B (N) of Γ 0 (N) in PSL(2, 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 Γ 0 (N) to obtain a chain, and investigated regular maps corresponding to the subgroups Γ 0 (N) 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 Γ and Γ(N).
The main purpose of the present paper is to describe a modulo-N subgroup of the normalizer Γ B (N) 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 Γ h (N). 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 Γ B (N) a triangular group. In this way we guarantee that all the maps are regular and we can reveal their arithmetic structure.

The Structure of the Normalizer and Some Subgroups
As described in [9], the normalizer Γ B (N) of Γ 0 (N) consists of the transformations corresponding to the matrices where all symbols represent integers, h is the largest divisor of 24 for which h 2 | 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 | 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 Γ h (N). It consists of the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | N.
Here we present a special subgroup of Γ(N) in order to calculate the index of Γ h (N) in Γ B (N). We denote this subgroup of Γ(N) by Γ * (N), which consists of the transformations corresponding to the matrices a bN cN d ∈ Γ(N) such that c ≡ 0 mod N/h 2 .
The following remark directly follows from Remark 1.
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.
where the resulting matrix is divided by e in order to obtain a matrix with determinant 1. Since This completes the proof.
Finally, using Theorem 1 and Remark 2, we can present the following Proposition.
where ρ is the number of distinct prime factors of N/h 2 and Proof. This is straightforward due to the relation The second important subgroup of Γ B (N) that we described is Γ h 1 (N), which consists of the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | N.
Proof. By definition, it is easily seen that Γ h (N) is the kernel of the homomorphism ψ : 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.
where ρ is the number of distinct prime factors of N/h 2 and The last subgroup of Γ B (N) that we described is Γ h B (N), which consists of the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | 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.
The resulting matrix has determinant e 2 . Thus, dividing all terms by e, we have a matrix with determinant 1 Furthermore, c ≡ 0 mod eh 2 and z ≡ 0 mod N yield Thus, using (3) and (4), we conclude . Then, for an arbitrary element where the resulting matrix has determinant e 2 . Dividing each element by e, we have Since the resulting matrix is in Γ h 1 (N), we have We now present two important theorems of [20] regarding the structure of Γ B (N). In these cases In the present study we investigate the regular 3-valent, 4-valent, and 6-valent maps corresponding to normal subgroups of Γ B (N).
Let us identify parameter N corresponding to each statement of Remark 3.  i.
According to the definition of N 1 , we have N 1 = h 2 and hence e = 1. Thus, Γ B (N 1 ) consists of the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | N 1 . ii. According to the definition of N 2 , we have N 2 = 2h 2 and hence e = 1, 2. Thus, Γ B (N 2 ) consists of two types of elements, namely, even elements and odd elements [19]. Even elements are the transformations corresponding to the matrices and odd elements are the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | N 2 . iii. According to definition of N 3 , we have N 2 = 3h 2 and hence e = 1, 3. Similarly to ii, Γ B (N 3 ) consists of even elements and odd elements [19]. Even elements are the transformations corresponding to the matrices and odd elements are the transformations corresponding to the matrices where h is the largest divisor of 24 for which h 2 | N 3 .

The Normalizer Maps
In this section, we construct three universal maps, namely, the normalizer maps M h 3 , M h 4 , and M h 6 . These universal maps are investigated in [18,19]. On the other hand, Akbaş showed that Γ B (N) acts transitively onQ for N 1 , N 2 , and N 3 . Now we are ready to construct the normalizer maps.    (resp. ad − 3bc = ±1). We know that M h 4 (resp. M h 6 ) is quadrilateral (resp. hexagonal), and its principal quadrilateral (resp. hexagon) is ∞, 0, ). One can find other quadrilaterals (resp. hexagons) applying the elements of Γ B (N 2 ) (resp. Γ B (N 3 )) 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 Γ(2, ∞, 3) and any regular triangular map is the quotient of the universal triangular map by a normal subgroup of Γ(2, ∞, 3). As Γ h (N) is a normal subgroup of Γ B (N) and Γ B (N) is a triangle group for all values of N 1 , N 2 , and N 3 , we can form the regular maps M h , we will call these regular maps "normalizer maps modulo N". Before investigating regular maps M h i (N), i = 3, 4, 6 analytically, we present some results concerning the normalizer Γ B (N) and its subgroups that we already described in Section 2.

Normalizer Maps Modulo
A straightforward calculation yields that the stabilizer S ∞ of ∞ in Γ B (N) is the cyclic group generated by 1 Consider the following matrix As a ≡ 1 mod N and c ≡ 0 mod N, we have a + cuN/h 2 ≡ 1 mod N. Furthermore, d ≡ 1 mod N yields that the resulting matrix is in Γ h 1 (N). For the converse, let T = a b/h cN/h d ∈ Γ h 1 (N). Consider the following equality Since a ≡ 1 mod N and c ≡ 0 mod N, a − abcN/h 2 ≡ 1 mod N. Furthermore, This completes the proof. Now we are ready to find the number of the vertices of M h i (N), i = 3, 4, 6. We give the theorem only for M h 3 (N 1 ). This theorem also holds for M h 4 (N 2 ) and M h 6 (N 2 ).  N 1 ). 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
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. Proof. If e 1 = e 2 then the determinant of T −1 2 T 1 cannot be 1 such that follows from the definition of Γ h 1 (N). Proposition 9 states that each e determines different types of vertices. For the map M h 3 (N 1 ), since e = 1, we have just one type of vertex. So we simply call them vertices. However, for M h 4 (N 2 ) (M h 6 (N 3 )), 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 Γ h 1 (N) corresponding to the matrix ae b/h cN/h de by the row vector (ae, cN/eh 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/eh 2 = 1, we have that (ae, cN/eh 2 ) = 1. So we denote the vertices of the maps by (ae, cN/eh 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 M h 3 (N 1 ), M h 4 (N 2 ), and M h 6 (N 3 ), using (5)-(9), we have the following cases. i.
For the case N 1 , since N 1 = h 2 , we have e = 1. Thus, using Proposition 9, the set of vertices of M h ii. 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 M h iii. 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 M h A directed edge of a map will be called a dart. Since Γ B (N) for all parameters N 1 , N 2 , and N 3 acts transitively onQ, it acts transitively on the darts of M h i for i = 3, 4, 6. Thus, Γ B (N)/Γ h (N) acts transitively on the darts of M h i (N) for i = 3, 4, 6 and N 1 , N 2 , N 3 because Γ h (N) is a normal subgroup of Γ B (N). According to the results of [16], a map is regular if its automorphism group acts transitively on its darts, which makes M h i (N) 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 M h 3 (N 1 ). Again, the theorem also holds for M h 4 (N 2 ) and M h 6 (N 3 ).

Theorem 5.
There exists a one-to-one correspondence between the left cosets of Γ h (N 1 ) in Γ B (N 1 ) and the darts of M h 3 (N 1 ).
Proof. Due to the transitivity of the action of Γ B (N 1 ) onQ and Γ h (N 1 ) The matrix on the right-hand side of Equation (10) must be in Γ h (N 1 ). According to the definition of Γ h (N 1 ) and Γ h 1 (N 1 ) we have a 1 a 2 + b 1 c 2 N 1 /h 2 ≡ c 1 b 2 N 1 /h 2 + d 1 d 2 ≡ 1 mod N 1 , and a 2 c 1 + c 2 d 1 ≡ 0 mod N 1 . Thus, a 1 b 2 + b 1 d 2 ≡ 0 mod N 1 must hold. Since b 2 ≡ 0 mod N 1 and S is arbitrary, we have b 1 ≡ 0 mod N 1 . This yields T ∈ Γ h (N 1 ). Again, the orbit-stabilizer theorem applies.
Based on Theorem 5 and Proposition 3, the following corollary follows. 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.