The Distant Graph of the Ring of Integers and Its Representations in the Modular Group

There is presented an infinite class of subgroups of the modular group $$\mathrm{PSL}(2,\mathbb {Z})$$PSL(2,Z) that serve as Cayley representations of the distant graph of the projective line of integers. They are infinite countable free products of subgroups of $$\mathrm{PSL}(2,\mathbb {Z})$$PSL(2,Z) isomorphic with $$\mathbb {Z}_2$$Z2, $$\mathbb {Z}_3$$Z3 and $$\mathbb {Z}$$Z subject to the restriction that the number of copies of $$\mathbb {Z}$$Z is 0 or 2. The proof technique is based on a 1–1 correspondence between some involutions $$\iota $$ι of $$\mathbb {Z}$$Z that fulfill the equation $$\begin{aligned} \iota (\iota (n)-\delta _n)=\iota (n+1)+\delta _{n+1},\quad \delta _n=\pm \,1, \quad \delta _{\iota (n)}=\delta _n, \end{aligned}$$ι(ι(n)-δn)=ι(n+1)+δn+1,δn=±1,δι(n)=δn,and groups from this class.


Introduction
Among many constructions of graphs corresponding to a ring in the case of integers the distant graph on the projective line P(Z) seems most suitable. It is also called the distant graph of the ring of integers. The properties of distant graphs have been studied in many papers, for instance [1][2][3]5,6]. According to the general definition of the distant graph (see i.e. [3]) the vertices are all cyclic submodules of the Z-modul Z 2 generated by vectors with co-prime coordinates. The edges of the graph connect vertices whose generators are the rows of an invertible (2 × 2)-matrix over Z. The graph is connected, has three elements maximal cliques and the group PGL(2, Z) as the automorphism group (we omit the proof of the latter fact since it is not crucial for our consideration).
In the paper [6] it was shown that the distant graph on P(Z) is a Cayley one. There where indicated its two non-isomorphic Cayley groups being subgroups of the modular group and presented a connections with Stern-Brocot tree. The question arises to describe all Cayley groups of this graph. Since the Cayley group of the graph is a subgroup of the automorphism group of this graph, these groups are subgroups of PGL(2, Z). The paper is a continuation of the last section of [6]. We prove that to any Cayley group of the distant graph on P(Z) an involution ι of Z corresponds satisfying the equation and give a recursive construction of a wide class of such involutions. Since we do not know any general method to construct such involutions, we restrict our attention to groups contained entirely in PSL(2, Z). The latter group we treat as the free product of groups PSL(2, Z) Z 3 * Z 2 with the presentation and I = ± 1 0 0 1 . Note that S n can be written as According to the Kurosh theorem any subgroup of PSL(2, Z) is a free product of some copies of Z 2 , Z 3 and Z. We prove that as a Cayley group of the distant graph on P(Z) we can take any infinite free product of these groups with the restriction that the number of copies of Z is 0 or 2.
In the proof we construct an appropriate involution ι and we use the argumentation of Marshall Hall Jr. from the paper [4] to reveal the form of {S n : n ∈ Z} . In a free product he introduced a so called semi-alphabetical order denoted by ≺. In the subgroup G of a group H he finds a set K = K(G) which consist of all nonidentical elements K ∈ G such that K does not belong to the group generated by elements that precede K in the semi-alphabetical order. By this proof the subgroup G is the free product of groups generated by elements of the set K. To find the set K for our groups we introduce three properties of subsets of PSL(2, Z).
We say that the set F of elements of the modular group has a property (G) if it contains no product of elements from F ∪ F −1 ; (P) if for any F j ∈ F and ε j ∈ {−1, Lemma 1.1. Let G be a subgroup of the modular group. If some F ⊂ G generates G and satisfies (G), (P) and (I) then F = K(G).
Since F generates G, each G j , hence G as well, is a non-trivial product of elements of F ∪ F −1 . We get G ∈ F, by the property (G).
On the other hand, if G ∈ G\F then either G = F −1 for some F ∈ F or G must be a product of some elements of F ∪ F −1 . Therefore G ∈ K, in the former case by the property (I) and in the latter case by the property (P).

Involutions of Z
Assume now that the distant graph on P(Z) has a Cayley group (G, S). Since our graph is vertex-transitive, we may assume that the neutral element of G corresponds to e = ± 1 0 . Its neighbourhood consists of v n = ± n 1 ∈ P(Z), It is not difficult to see that the automorphism group of our distant graph is isomorphic to PGL(2, Z). This fact allows us to treat elements of G as members of this group. As S = S −1 we have S −1 n = S ι(n) for some bijection ι : If S n = ± a b c d , from the above equation applied to n and ι(n) we get are the only candidates for members of S. Straightforward calculations show that (2.1) holds iff ι satisfies with δ n = detS n . In the paper [6] we found two non-isomorphic Cayley groups G of the distant graph on P(Z) satisfying the additional condition In the very end we will use the following observation. The proof of the lemma in one direction is above and in the other follows from the considerations after Lemma 4.3 of [6] (at that point we use (2.3)). Therefore, in order to construct a Cayley group of the distant graph on P(Z) it is enough to find an involution of Z satisfying (2.2). For this we define some type of involutions on finite sets of consecutive integers. A mapι : 2) for each n = k, . . . , k + l − 1. We can freely shift the domain of a building involution along integers. We now present a recursive method of constructing required involutions. . . , k n +l n } −→ {k n , . . . , k n + l n }, n ∈ N satisfying k 0 = −1, k 1 = l 0 , k n+1 = k n + l n + 2 for n ≥ 1. We define an involution ι = ∞ n=0 ι n : Z −→ Z as a "limit" construction: ι 0 , ι 0 ι 1 , ι 0 ι 1 ι 2 , . . . , ι 0 ι 1 ι 2 . . . ι n , . . . (2.4) i.e. we require that ι| {−n−1,...,kn+ln+1} = ι 0 ι 1 ι 2 . . . ι n for each n ≥ 1. It follows immediately from the construction that ι thus defined satisfies (2.2).
It is possible to find an involution of Z satisfying (2.2), with δ n = −1 for some n ∈ Z.

Constructions of Cayley Representations
Given p ∈ N let us denote by F p , T p and D p the free group with p free generators, the free product of p copies of Z 3 and the free product of p copies of Z 2 , respectively. With such notations we have the following result.
is a Cayley group of the distant graph on P(Z).
Proof. First we deal with the case p = 0. Observe that if l n ∈ {0, 1} then ι n is uniquely determined. Moreover if l n = 0 then S kn is an involution in PSL(2, Z), and if l n = 1 then S kn is of order 3. It follows that in the case we deal with in order to define ι it is enough to choose a sequence (l n ) ∞ n=0 ∈ {0, 1} N . Put then 1. q = 0: l n ≡ 0; 2. 0 < q < ℵ 0 : 3. r = 0: l n ≡ 1; 4. 0 < r < ℵ 0 : l n = ⎧ ⎨ ⎩ 0 : 1 ≤ n ≤ r, 1 : n = 0 or n > r 5. q = r = ℵ 0 : l n = 1 2 (1 + (−1) n ). We have defined the set S. In order to show that the generated group G = S is an appropriate one we have to change the set of generators to obtain one with the properties (G), (P) and (I). Put , n ≥ 1 and denote F = {F n : n ≥ 0}. Since ι(k n − 1) = −n + 1 − l 0 for n ≥ 1, and ι(k n ) = k n + l n for n ≥ 0,

Results Math
It follows that the set F generates G since l n ∈ {0, 1}. For n ≥ 1 we have F n in the reducible form Recall that l 0 = 0 iff q = 0 iff l n ≡ 0.
By inspection one can check that F satisfies the property (I). Properties (G) and (P) of the set F for the case 1. (and 2. with q = 1) are shown in [6], hence assume that l 0 = 1 (i.e. we consider cases 2. -5.).
Observe that both F n and F −1 n , n ≥ 1, are of the irreducible form By easy induction we get that an arbitrary product F ε1 n1 ·. . .·F εt nt , where ε j = ±1 and n j = 0, j = 1, . . . , t, is of the irreducible form (3.5) The above formula allows us to show the property (P) of F.
First observe that trivially we have If we assume that n j = 0, j = 1, . . . , t, then in virtue of (3.5) and (3.4) we have Now we deal with the case p = 2.
The irreducible form of F n is the following Again by inspection one can check that F satisfies the property (I).
Before we proceed to the proof of property (P) denote N 1 = {0, 1}, N 2 = {2, 3, . . .}, F j = {F n : n ∈ N j }, j = 1, 2, and note that (3.8) In virtue of 1. of (3.8) it is enough to show the property (P) separately for the sets F 1 and F 2 . Let G = F ε1 n1 · . . . · F εt nt . In the former case from 2. of (3.8) and the first part of (3.7) it follows that In the latter case from 3. of (3.8) and the second part of (3.7) it follows that Considerations similar to those in the previous case convince us that the property (G) is satisfied. Therefore Lemma 1.1 tells us that F = K(G) is satisfied in each considered case. Now from Hall's proof of the Kurosh subgroup theorem it follows that G F p * T q * D r since, in virtue of (3.3), (3.6) and the definition of (l n ), the set F consists of an appropriate number of elements of infinite order, elements of order three and involutions.
From (3.3) and (3.6) it immediately follows that (2.3) is satisfied in all cases. In virtue of Lemma 2.1 each of (G, S) is a Cayley representation of the distant graph on P(Z).
We are not able to find a building involution that produces an odd number of independent generators of infinite order. Taking this into account and based on the above proof, we make the following conjecture. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.