A Note on Transitive Action of the Extended Modular Group on Rational Numbers

: The extended modular group Γ is isomorphic to the amalgamated free product of two dihedral groups D 2 and D 3 with amalgamation Z 2 . This group acts on rational numbers transitively. In this study, we obtain elements in the extended modular group that are mappings between given two rationals. Also, we express these elements as a word in generators. We use interesting relations between continued fractions and the Farey graph.


Introduction
The modular group Γ = P SL ( By taking S = T U = ( 0 −1 1 1 ), the presentation of Γ is Let us denote the set G = {( a b c d ) ∶ a, b, c, d ∈ Z, ad − bc = −1}. The corresponding transformations of elements in G are anti-automorphisms. Thus, the extended modular group can be defined as Γ = P SL(2, Z) ∪ G . The extended modular group is isomorphic to free product of two dihedral groups of order 4 and 6 , amalgamated with a cyclic group of order 2, i.e., where R = ( 0 1 1 0 ) is a reflection map.
In this study, we focus on the action of the extended modular group Γ on rational numbers.
Every rational number has a reduced fraction p q = −p −q , where p, q ∈ Z and (p, q) = 1. We represent ∞ as 1 0 = −1 0 . Consider the element V = ( a b c d ) ∈ Γ and the corresponding Möbius transformation Here ap+bq cp+dq is also a reduced fraction. Additionally, the Diophantine equation As a result, the action of the extended modular group on rationals is transitive [14].
Our aim is to find an element V ∈ Γ for given two rationals p represent V as a word in generators.

Motivation and Background Materials
In this section, we give some information about continued fractions, the Farey sequence, the Farey graph and relations to the extended modular group. For more information see [1,2,6].
There are impressive relations between the modular group and continued fractions. Let where r j ∈ Z and i = 0, 1. The corresponding Möbius transformation of this element is In addition, the image of infinity is a continued fraction expansion of a c . This expansion is the Rosen continued fraction defined in [11] for λ = 1, and it is called integer continued fraction expansion.
In this expansion, for i ≤ n , C i = pi qi = [r 0 ; r 1 , ..., r i ] is called ith convergent of the expansion.
It can be seen by calculation On the other hand, it is possible to make connections between integer continued fractions and the Farey sequence.
The Farey sequence of order n is a complete and ordered set of reduced rational numbers in the interval [0, 1] which have denominators less than or equal to n .
It can be seen that if a c and b d appears one after another in some F n , then ad − bc = ±1. We called such rationals Farey neighbours. All Farey neighbours of a rational x is denoted by N (x). The Farey sum of a c and b d defined as, Observe that every F n includes F n−1 and new members are obtained by Farey sum of its neighbours. For instance 1 2 ∈ F 2 is the Farey sum of 0 1 and 1 1 in F 1 . This rule is known as the mediant rule. It should be noted that if the denominator of Farey sum of two neighbours in F n−1 exceeds n , then this rational number will not appear in F n since the definition of the Farey sequence. Definition of the Farey sequence can be extended toQ = Q ∪ {∞} by assuming ∞ = 1 0 . Hence, for a given rational a c , it is known that a c has finite integer continued fraction expansion.
In addition, b d is the penultimate convergent of the integer continued fraction expansion of a c .
This yields ad − bc = ±1 , in other words, a c and b d are Farey neighbours. As a result, (  Figure 1. In recent years, many studies have contributed the continued fractions related to the action of some subgroups Möbius transformations. In [2], integer continued fraction expansions and geodesic expansions are studied from the perspective of graph theory. Short and Walker used Rosen continued fractions as paths in a class of graphs in hyperbolic geometry [13]. Same authors also studied connections between even integer continued fractions and the Farey graph [12]. Relations between cusp points and Fibonacci numbers are studied in [7] using Farey graph and continued fractions. Algebraic and combinatorial properties of continued fractions and modular group related with Farey graph are given in [10]. Besides that some relations between elliptic elements and circuits in graph for normalisers of subgroups of P SL(2, R) are examined in [4,5].

Main Results
Firstly, we obtain matrix representation of the elements in Γ which the corresponding transformation is a mapping between given two rationals. For a given reduced rational x = p q and a neighbour In other words, the corresponding transformation pz+r qz+s is an automorphism. In this case, it is possible to construct an anti-automorphism element by taking −r for r and −s for s , i.e., ( p −r q −s ). Similar observations can be done for the case y > x . For convenience throughout this paper, we need to define a location function µ x ∶ N (x) → {−1, +1} for neighbours of a rational: Now we are ready to obtain a mapping between two rationals.
Then the corresponding transformation of the element The equality µ p q ( r s maps p q to p ′ q ′ . Furthermore, V is a reflection.

Proof
Since p q and p ′ q ′ are adjacent in the Farey graph, we can take V 1 = ( p p ′ q q ′ ) and Proof follows similar to the proof of Lemma 3.1. It can be seen easily that This equality proves that V is an anti-automorphism. It is obvious that V is a reflection since T r(V ) = 0. ◻ Considering the arguments mentioned before Lemma 3.1, we can map p q to p ′ q ′ via an elliptic element. It is enough to take V 2 = ( p ′ −p q ′ −q ) in the proof of Corolary 3.2. Therefore, we omit the proof of the following corollary.
Now our aim is to obtain a generalization of Lemma 3.1. For doing this, we need more information about Farey neighbours. As we mentioned in the motivation section, the Farey sequence of level n is a complete and ordered set of reduced rationals which have denominators less than or equal to n . Every F n includes F n−1 . New members obtained via mediant rule. More clearly, if a c and b d is contained in F n−1 , then the mediant of these two terms a c ⊕ b d = a+b c+d is contained in F n on one condition that c + d ≤ n . If a reduced rational p q first appears in F n via Farey sum of a c and b d in F n−1 , then a c and b d is called Farey parents of p q . After that all Farey neighbours of p q will be of the form, A basic result of this, is the following lemma.

Lemma 3.4
Let p q be a reduced rational number and r s , r ′ s ′ ∈ N ( p q ). Then, there exists an integer k such that Here we consider the element ( p r q s ) ∈ Γ as an automorphism (or anti-automorphism).
Using Lemma 3.4, we can construct another automorphism (anti-automorphism) element with cusp point p q .
Lemma 3.5 Let V 1 , V 2 ∈ Γ with same cusp point. Then there exists an integer k such that Here U = ( 1 1 0 1 ) is the parabolic generator of Γ .
Proof Suppose the common cusp point is p q . For neighbours r s , r ′ s ′ ∈ N ( p q ), we can think V 1 and V 2 as From Lemma 3.4, we have an integer k such that Hence, we complete the proof by considering the generator U = ( The above lemma tells us V 1 and V 1 .U k have common cusp point for every integer k , and that is the key for the following theorem which is a generalization of Lemma 3.1.
. Then for every k ∈ Z , the corresponding transformation of the element maps the rational p q to p ′ q ′ .
Proof We use a similar technique of the proof of Lemma 3.1. From Lemma 3.5, the elements ( have cusp points p q and p ′ q ′ , respectively, for k 1 , k 2 ∈ Z . Then,

Words in Generators
In this section, we consider relations between Farey paths and integer continued fractions. Using these relations, we obtain extended modular group elements as words in terms of generators that correspond a transformation between given two rationals.
for every integer k . In addition, the anti-automorphism has the word form Proof First we map p q to 0. Considering the equality (1), The two ordered elliptic generator T maps 0 to infinity. Then, the parabolic generator U fixes infinity. Finally, the cusp point of the element is p ′ q ′ which proves the result. The second part of the proof can be done by considering the element Since the modular group is isomorphic to the free product of the cyclic groups of orders 2 and 3, every element can be expressed as a word in T and S . Considering U = T S , we get two blocks, Hence, every element V ∈ Γ has a word form, where i = 0, 1, 2 , j, t = 0, 1 . The powers of the blocks are positive integers but m 0 and n k may be zero. This form is called block reduced form. Every element in the extended modular group has block reduced form. For instance, RT S 2 RT ST S 2 T S 2 RT S can be expressed as (T S) 2 .(T S 2 ) 3 R .
Trace classes of the modular group and the extended modular group were studied in [3,8].
Corresponding transformations of these blocks are related to simple continued fraction expansions.
Here we obtain block forms of the words given in Theorem 4.1.

Theorem 4.2
The block reduced form of the elements given in (2) and (3) are respectively.
Since the elliptic generator T is of order 2 and S is of order 3, we have the block reduced form of the word as stated. The second part of the proof can be obtained similarly with relations ◻ Before we sum up all our results, we make connections with Farey paths. A path in a graph consists of consecutive adjacent vertices. So, a Farey path < v 1 , v 2 , ..., v n > is a path such that v i = pi qi for i = 1, 2, ..., n are reduced rationals and since the consecutive v i 's are adjacent, we have As Farey graph is connected, there always be a path between two rationals.
For a given reduced rational p q = [r 0 ; r 1 , ..., r n ], the ith convergent of the integer continued fraction expansion of p q defined as C i = pi qi = [r 0 ; r 1 , ..., r i ] for 0 ≤ i ≤ n , where C 0 = p0 q0 = r0 1 and C n = pn qn = p q . Furthermore, we know that p i .q i−1 − q i .p i−1 = ±1 . Hence, every consecutive pair C i and C i−1 are Farey neighbours. Also, this situation can be thought as < ∞, C 0 , C 1 , ..., C n−1 , C n > is a path from ∞ to p q . Finally, every integer continued fraction expansion of a rational p q is The elements V and V 1 have the word forms W and W ′ for k = 0, respectively. Finally, we express W and W ′ in block reduced forms by Theorem 4.2,

Conclusion
For given two reduced rationals p q and p ′ q ′ , we obtain elements V ∈ Γ such that V ( p q ) = p ′ q ′ . We use the relations between paths in the Farey graph and continued fractions to get V as a word in terms of generators. We also obtain the block reduced form that is a word contains only finite ordered elements. For future research one can consider the blocks, f = RT S = ( 0 1 1 1 ) and h = T SR = ( 1 1 1 0 ) defined in [9]. Powers of these matrices have only Fibonacci entries. Koruoğlu proved that every element can be written as a word in powers of f and h . This word is called New Block Reduced Form [9]. Obtaining new block reduced form of the words given in this study, makes relations to the Fibonacci sequence.

Declaration of Ethical Standards
The author declares that the materials and methods used in their study do not require ethical committee and/or legal special permission.