Number of Distinct Fragments in Coset Diagrams for $PSL\left(2,\mathbb{Z}\right)$

Abstract

Coset diagrams [1, 2] are used to demonstrate the graphical representation of the action of the extended modular group P G L 2 , Z over P L F q = F q ⋃ ∞ . In these sorts of graphs, a closed path of edges and triangles is known as a circuit, and a fragment is emerged by the connection of two or more circuits. The coset diagram evolves through the joining of these fragments. If one vertex of the circuit is fixed by a x ρ 1 a x − 1 ρ 2 a x ρ 3 ⋯ a x − 1 ρ k ∈ P S L 2 , Z , then this circuit is termed to be a length – k circuit, denoted by ρ 1 , ρ 2 , ρ 3 , ⋯ , ρ k . In this study, we consider two circuits of length − 6 as Ω 1 = α 1 , α 2 , α 3 , α 4 , α 5 , α 6 and Ω 2 = β 1 , β 2 , β 3 , β 4 , β 5 , β 6 with the vertical axis of symmetry that is α 2 = α 6 , α 3 = α 5 and β 2 = β 6 , β 3 = β 5 . It is supposed that Ω is a fragment formed by joining Ω 1 and Ω 2 at a certain point. The condition for existence of a fragment is given in [3] in the form of a polynomial in Z z . If we change the pair of vertices and connect them, then the resulting fragment and the fragment Ω may coincide. In this article, we find the total number of distinct fragments by joining all the vertices of Ω 1 with the vertices of Ω 2 provided the condition β 4 < β 3 < β 2 < β 1 < α 4 < α 3 < α 2 < α 1 .