Evolution of ambiguous numbers under the actions of a Bianchi group

In this paper we study some combinatorial properties of biquadratic irrational number field , under the action of a Bianchi group . In this experiment it is revealed that a special class of elements exists; that is, for an element its conjugate has different signs in the closed path (orbits) for the action of over , known as ambiguous numbers. It is also proved that the orbit defined on a finite number of ambiguous numbers succeeding a unique closed path.


Introduction
A significant portion of the combinatorial group theory is about exploring the subgroups of projective special linear group over the ring of complex numbers, that is, PSL (2, C). The study of PSL(2, C) (or PGL (2, C)) comprising all Linear Fractional Transformations (LFTs), with complex coefficients, was one of the mainstream topics of mathematics in the last century and played an important role in the development of Lobachevskian geometry (Non-Euclidean geometry). A special class of discrete subgroups of projective special linear groups PSL (2, C) are the groups of the form PSL (2, O r ), where O r is the ring of integers in the imaginary quadratic irrational number field Q( √ −r). The ring of integers O r has a Euclidean algorithm only when r ∈ {1, 2, 3, 7, 11} while only two rings of integers for r = 1, 2; that is, O 1 and O 3 have non-trivial units ( = ±1). The groups r = PSL(2, O r ) with r and O r as above are known as Bianchi groups (see [1][2][3][4][5][6][7]). The Bianchi group 3 = PSL (2, O 3 ) can be represented finitely with three generators that satisfy seven relations. The LFTs concerned to three generators x, y and t are x : z → − 1 z , y : z → w −1 z w and t : z → z−1 z , where w = −1+ √ 3i 2 . In [8], the finite presentation of 3 is formed by adjoining √ k 1 and √ k 2 , where k 1 and k 2 are square-free integers, is called biquadratic field over Q [9]. The elements of the field Q √ k 1 , √ k 2 are of the form: It is known that 3 acts on Q i, √ r , where r is a positive square-free [9][10][11]. The generators x, y and t of 3 have fixed points ±i, 0 and 1 ± √ 3i/2, respectively. All fixed points are placed in a biquadratic field Q(i, √ 3), where √ 3 and i are zeroes of an irreducible ring of polynomial, that is, (q 2 − 3)(q 2 + 1) over Q, for more detail [10,12,13]. The action of 3 on Q(i, √ 3) deserves special treatment because Q(i, √ 3) has all the fixed points of generators of 3 and these actions are also differentiated from Q i, √ r , where r is a positive square-free [10]. Mushtaq in [14] defined a coset diagram for the modular group PSL(2, Z) and after that many authors used the coset graph to study different group theoretic properties, while considering the action on certain base fields accordingly, for details see [9,10,[12][13][14][15][16][17][18][19][20][21] and some related number theoretic applications in [22,23]. The elements of Q(i, . Therefore, these elements deserve a special kind of classification. There always exist two conjugates [24], namely ξ = α+β has also a conjugate of ξ again, so we have a conjugate of ξ over irrational number field 3) is a real quadratic irrational number, if ξ andξ are both positive (or both negative), where α, β, and γ ∈ Q are said to be a completely positive (or completely negative). In [13,19], Mushtaq discussed and defined a special type of numbers known as ambiguous numbers and proved that an ambiguous number exists if ξ and its conjugateξ have opposite signs. The action of Bianchi group 3 has played a very important role in the classification of the orbits of Q(i, √ 3). For detailed results and discussion related to Bianchi groups readers referred to [1][2][3]5,10,15,[25][26][27][28][29][30][31][32][33]. It is obvious to see the application of group theory to mechanics and physics to construct models, drive differential equations and investigate their structures [34][35][36].
The major contributions of this work are listed below.
(i) This paper presents a novel graphical study of the action of Bianchi group PSL(2, O 3 ) on the biquadratic irrational field. (ii) We have discovered a new class of elements of the bi-quadratic irrational field, possessing some interesting properties, known as ambiguous numbers. (iii) We proved that ambiguous numbers in the coset diagram form one and only one closed path (orbit) for ξ .

Action of PSL(2, O
We have clarified how ambiguous numbers would create a path from one ambiguous number to the next in the following proposition.

Lemma 2.3:
is completely a positive (negative) real quadratic irrational number, then t(ξ ) is completely positive and t 2 (ξ ) is completely a negative number.

Lemma 2.4: Let 3 act on Q(I, √
3), then the transformation x maps one ambiguous number to another.

Lemma 2.5: If x(ξ )
is an ambiguous number, then ξ is also an ambiguous number.
. Therefore, y(ξ ) is an ambiguous only if the imaginary part of the equation is zero, that is = 0. The real part will also be equal zero if α = β = γ = 0. Therefore, ξ = ∞ is not an ambiguous number, that's why β and γ cannot be zero. This proves that y(ξ ) is not an ambiguous number. Also, y 2 is an imaginary part of this equation and hence y 2 (ξ ) is an ambiguous number only if (α = 0. If α, β and γ are zero, then the real part of the equation will also be equal to zero. Hence ξ = ∞ is not an ambiguous number, that's why β and γ cannot be zero. Hence, y 2 (ξ ) is not an ambiguous number. is an ambiguous number and t(ξ ) is completely a positive number.
x(2. 16 It is clear from the above discussion 1-23 and Figure 1. if we have a positive ambiguous number ξ , then the transformation t (ξ) is also a completely positive number and t 2 (ξ ) is an ambiguous number. If ξ is a negative ambiguous number, then the transformation t 2 (ξ ) is also a completely negative number and t(ξ) is an ambiguous number, by theorem 2.4, generator x is used to join these ambiguous numbers to another ambiguous numbers.
Through inductive hypothesis, as demonstrated in Figure 1 and by virtue of lemma 2.10, there exist finite ambiguous numbers. Now, if we start from one vertex, that is an ambiguous number ξ n 2 (superscript in ξ n 2 is pointed as the number of triangle and subscript is pointed as the number of vertex of triangle labelled by ambiguous numbers) n is an odd number; after a finite number of steps, that is, x(ξ n 2 ) = ξ 1 1 , because the generator x maps one ambiguous number to the next. Hence, there exists a sequence, ξ = ξ 1 1 , ξ 1 2 , ξ 2 1 , ξ 2 2 , . . . ξ n 1 , ξ n 2 , ξ 1 1 = ξ of ambiguous numbers that forms a unique closed path.

Conclusion
In this work, we have discussed group theoretical aspects of the actions of a Bianchi group 3 on Q(i, √ 3). Since the closed path can be defined as the path where the vertices of the initial and the terminal (end) coincide, the closed path of ambiguous numbers as a closed path with all ambiguous numbers at its vertices. We have proved that for the orbit ξ , there exist a finite number of ambiguous numbers, where they form a unique closed path.