On Suborbital Graphs for the Group $$\varGamma ^{3}$$

Normalizer Maps Modulo N

“…For N 3 = 3, we have h = 1 and the regular map M 1 6 (3). The numbers of vertices, edges, , and faces are V (1, 0), (1,3), (1,6). This is the map {6, 3} (0,2) lying on the torus U /Γ 1 (3) with Schlafli {6, 3} (see Figure 4).…”

“…Hence, we can regard the elements of PSL(2, R) as the matrices 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].…”

Petrie Paths in Triangular Normalizer Maps

“…The normalizer of the congruence subgroups Γ 0 (N) of the modular group Γ in PSL(2, R) is Γ B (N) which is simply called "the normalizer". We refer reader to [2][3][4][5][6][7] and references therein for results concerning the normalizer. It is known that the normalizer is a triangle group, and it acts transitively on the set of extended rational numbers Q = Q ∪ {∞} for N = 2 α 3 β with α ∈ {0, 2, 4, 6} and β ∈ {0, 2} [8,9].…”