Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

Abstract

In this study, we work on the Fuchsian group where m is a prime number acting on transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

1. Introduction

The Hecke group, , introduced by Hecke in [1], is the group generated by the two Möbius transformationswhere λ is a real number such that and q is an integer greater than 2 or . When , is the modular group . If , it is known, see [2], that , where consists of all transformations of the following two types:

However, Rosen [3] showed that the above two transformations need not to be in if . He proved that T is an element of if and only if is a finite λ-fraction, i.e.,where is an integer and is a positive integer for . Later, Keskin [4] presented the Fuchsian group for a squarefree positive integer m, which consists of all mappings of the forms (2) and (3).

In 1991, Jones et al. [5] studied the modular group by applying the idea of the suborbital graphs for a permutation group, introduced in [6]. Later, Akbas [7] proved his conjecture stating that a suborbital graph for the modular group is a forest if and only if it contains no triangles. They completely characterized circuits in suborbital graphs for the modular group. These studies lead us to explore infinite paths in suborbital graphs for the Fuchsian group .

In [8], Yayenie gave a remark stating that the Fuchsian group acts transitively on the set if and only if m is either 1 or prime. So, we study on the Fuchsian group where m is a prime number. We divide this work into four sections. In Section 2, we will show the way to construct a suborbital graph for the group and we give the edge conditions for two vertices that are joined in the graph. In Sections 3 and 4, we investigate vertices on the infinite path of minimal length in the graph for two cases: and , respectively; here, ordered pairs denote greatest common divisors. Finally, we represent each vertex on the infinite path of minimal length by recurrence relations and determine the limit point of the sequence of the vertices.

2. Suborbital Graphs for

Let and m be a squarefree positive integer, every element of can be represented as a fraction with and . We represent as . acts on naturally by

The following remark was given by Yayanie in [8], regarding the Fuchsian group acting transitively on vertices .

Remark 1. [8]. The Fuchsian group acts transitively on the set if and only if m is either 1 or prime.
From now on, it will be assumed that m is prime. So, acts transitively on . We will give a construction of suborbital graphs for the Fuchsian group . Let act on by. The orbits of this action are called suborbitals of . The suborbital containing is denoted by . We can form a suborbital graph whose vertices are the elements of , and there is a directed edge from λ to δ if , denoted by . We can see that is also a suborbital such that or . In the latter case, is just with reversed arrows and we call and paired suborbital graphs. In the case , the graph consists of pairs of oppositely directed edges, and we replace each pair with an undirected edge for convenience. We call the graph self-paired.
Since acts on transitively, each suborbital contains a pair for some . The following two theorems are valid for prime number m and we can use the same technique in the proofs of Theorem 1 and 2 in [9], which were stated for and 3.

Theorem 1. Let u and be relatively prime and m prime. If , then, there exists a directed edge from to in if and only if and either.(i) and or(ii) and .

Theorem 2. Let u and be relatively prime and m prime. If , then, there exists a directed edge from to in if and only if either.(i) and or(ii) and .Here, the choice of signs for x and y are always the same. By using Theorems 1 and 2, we obtain the following two corollaries that characterize a self-paired graph.

Corollary 1. Let u and be relatively prime and m prime such that . Then the suborbital graph is self-paired if and only if .

Corollary 2. Let u and be relatively prime and m prime such that . Then the suborbital graph is self-paired if and only if .
Next, we will show the existence of an integer k such that .

Lemma 1. Let u and be relatively prime and m prime such that . Then there exist integers k and l with such that and .

Proof. Since , we have . Then, there exists an integer x such that . So Taking , it is seen that is satisfied. Note that k and l are uniquely determined.

Theorem 3. Let u and be relatively prime and m prime such that . Suppose that and where are integers such that . If is self-paired, then ; otherwise, .

Proof. Since and , . As , we obtain that . So, there exists an integer y such that and then since . Hence, or 2. Assume that is self-paired. By Corollary 1, we have which implies that and . Since , , and . From , we get , so . For the case , we have .

Lemma 2. Let u and be relatively prime and m prime. Then there exist integers k and l with such that and .

Theorem 4. Let u and be relatively prime and m prime. Suppose that and where are integers such that . If is self-paired, then ; otherwise, .

3. Infinite Path of Minimal Length in Where  = 1

Let be vertices of the suborbital graph , we call the configurationsa path and an infinite path, respectively. If (or ) and there is no vertex which has greater (or smaller) value than joined with the vertex , then is the farthest vertex which can be joined with the vertex . The path is called of minimal length if and only if , where and must be the farthest vertex which can be joined with the vertex .

In this section, we focus on the infinite path of minimal length in the suborbital graph where . By the choice of prime, m, and remark 1, we obtain transitivity. Thus, we can map the first edge of any infinite path to the edge . We start investigating vertices in the infinite path of minimal length by determining the farthest vertex which can be joined with the vertex .

Theorem 5. Let u and be relatively prime and m prime such that , and let be the integers uniquely determined in Theorem 6. Then, we have the following results in :(i)The farthest vertices which can be joined with on the right and the left are respectively. No nearest vertex exists.(ii)The farthest vertices which can be joined with and are respectively. No nearest vertex exists.(iii)The farthest vertices which can be joined with and arerespectively. No nearest vertex exists.

Proof. For the right side of , we assume that there exists an edge in and . We can write in the formWith this and the fact that , we can replace withwhere is in . Let d be the greatest common divisor of and ; then, we get andTheorem 1 gives the conditions when this edge exists. Since , we have so case in Theorem 1 cannot happen. Then, we thus consider case . In this case, we have and .
If and , then which implies . As , we have , that is, . Since , we get . In other words, for some integer z. Thus,We will find the largest value of by defining a function ,The derivative of f is , which is negative for every nonnegative z. This implies that the maximum occurs at and maximum value isBy Theorem 1, it now suffices to show that is an irreducible fraction. Thus,is a vertex in and is the farthest one joined with . We also see thatThis implies that there is no such nearest point joined with the vertex .
If and , then , which implies . We have . This implies that ; that is, . The fact that implies that . Therefore, for some z in . Hence,The proof is similar to the previous case. Next, we will consider the left side of . Assume that there exists an edgein and . We can replace withwhere is in . Let c be the greatest common divisor of and ; then, we get andBy Theorem 1, case cannot happen. So, we will consider case . Then, we have and .
If and , then which implies . As , we have ; that is, . Since , we get . In other words, for some integer z. Thus,We define a function ,The derivative of f is , which is positive for every nonnegative z. This implies that the minimum occurs at and minimum value isBy Theorem 1, it now suffices to show that is an irreducible fraction. Thus,is a vertex in and is the farthest one joined with . We also see thatThis implies that there is no such nearest point joined with the vertex .
If and , then which implies . We have . This implies that ; that is, . The fact that implies that . Therefore, for some z in . Hence,This case is done by using a similar argument to that of the previous case.

Corollary 3. Let u and be relatively prime and m prime such that . If there are integers such that and , thenare elements of . Moreover, are the farthest vertices which can be joined with and , respectively, where .

Corollary 4. Let u and be relatively prime and m prime. If where , then there is an infinite path of minimal length:whose vertices are in the set

Corollary 5. Let u and be relatively prime and m prime. If where , then there is an infinite path of minimal length:whose vertices are in the set

4. Infinite Path of Minimal Length in Where

In the previous section, we provided the existence of infinite path of minimal length in the suborbital graph where . We find that the existence property is also valid for the suborbital graph where in very close analogy.

Theorem 6. Let u and be relatively prime and m prime such that , and let be the integers uniquely determined in Theorem 8. Then we have the following results in :(i)The farthest vertices which can be joined with on the right and the left are respectively. No nearest vertex exists.(ii)The farthest vertices which can be joined with and are respectively. No nearest vertex exists.(iii)The farthest vertices which can be joined with and arerespectively. No nearest vertex exists.

Corollary 6. Let u and be relatively prime and m prime such that . If there are integers such that and , thenare elements of . Moreover, are the farthest vertices which can be joined with and , respectively, where.

Corollary 7. Let u and be relatively prime and m prime. If where , then there is an infinite path of minimal length:whose vertices are in the set

Corollary 8. Let u and be relatively prime and m prime. If where , then there is an infinite path of minimal length:whose vertices are in the set

5. Continued Fractions and Recurrence Relations

From results in Sections 3 and 4, we have that any vertex on the infinite path of minimal length can be represented by a continued fraction expansion. As a continued fraction is related to recurrence relations, we use them to investigate vertices on the infinite path of minimal length. We conclude this section by finding the limit point of the sequence of the vertices.

Let be sequences of complex numbers with for and be sequences of Möbius transformations defined as follows:

We consider and so on and form a continued fraction of the form

For convenience, we denote this by

In [10], the numerator and the denominator of a continued fraction as in (43) are defined by the recurrence relationswith initial conditions

For a given sequence , can be written asand then

Now we consider infinite paths in suborbital graph . For the case when , the infinite path of minimal length for the right direction in Corollary 4 gives and for . By recurrence relations in (45), we obtain , and then, we have a vertex on this path:

Similarly, we have a vertex on the infinite path of minimal length for the left direction is

Theorem 7. If and , then we have

Proof. From the recurrence relation, we havewith and . The characteristic equation for the relation (53) iswhich gives two rootsThen, any solution of (9) have the formBy using the initial conditions, we havewhich impliesHence, we getSince is equal tothen we obtain

Theorem 8. If and , then we haveNext, we consider case . By Corollary 7, we have for . So we get from recurrence relations in (45). Then, a vertex on the infinite path of minimal length for the right direction isLikewise, a vertex on the infinite path of minimal length for the left direction is

Theorem 9. If and , then we have

Theorem 10. If and , then we haveHaving characterized the vertices on the infinite path of minimal length, we investigate the limit point of this path by using the Śleszyński–Pringsheim theorem.

Theorem 11. [10] (Śleszyński–Pringsheim). The continued fractionconverges to some valued f with if

Corollary 12. The sequence of the vertices of infinite path of minimal length (30) converges to

Proof. Since we have and where m is prime and , we get for all . By Theorem 11, the continued fractionconverges to f with ; that is, As we know, and since , we have Hence, andAs and , we get . Therefore, we obtain that the sequence of the vertices of infinite path of minimal length (30) converges to

Corollary 13. The sequence of the vertices of infinite path of minimal length (32) converges to

Corollary 14. The sequence of the vertices of infinite path of minimal length (38) converges to

Corollary 15. The sequence of the vertices of infinite path of minimal length (40) converges toWe observe that the limit points in Corollaries 23 and 24 are not in the set , but the limit points in Corollaries 25 and 26 will be in the set 2 if .

Data Availability

There are no data for supporting this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Chiang Mai University.