The Cayley Graph of Neumann Subgroups

All Cayley representations of the distant graph $\Gamma _Z$ over integers are characterized as Neumann subgroups of the extended modular group. Possible structures of Neumann subgroups are revealed and it is shown that every such a structure can be realized.


Introduction
In 1932 Neumann ([14]) investigated subgroups N * of the homogenous modular group SL(2, Z) which are defined by the condition (N ). (N ) For any ordered pair of relative prime integers (a, c) N * contains exactly one matrix in which the first column consist of the ordered par (a, c) Neumann investigated these subgroups in connection with problems of foundation of geometry. In 1973 Magnus explored subgroups N of the modular group M such the its natural extension N * by the central element of SL(2, Z) has Neumann (N ). He proved they are maximal nonparabolic subgroups of the modular group M ( [9]).
In [2], [3], [4] it was considered the notion of the distant graph over a ring with the identity, which is an combinatoric object represented the projective line P(R) over a ring R [1], [2]. In the case of the integers the vertices as the elements of P(Z) Q ∪ {∞} =:Q are all cyclic submodules of the Zmodule Z 2 generated by the vectors with co-prime coordinates. The edges of this graph connect vertices whose generators are the rows of an invertible (2 × 2)-matrix over Z. This distant graph we will denote by Γ Z . The graph Γ Z is depicted in Fig. 1. Note that we can construct this graph using the Stern-Brocot procedure twice. For the vectors with positive slopes start from [1,0] and [0, 1] and for the vectors with negative slopes from [1,0] and [0, −1]. To get Γ Z one has to just "glue" the vectors [0, 1] and [0, −1].
In [11] it was proved that the distant graph Γ Z is a Cayley one and then in [12] there were constructed uncountably many its Cayley representation. Inherently, it was proven that its Cayley representations in the modular group are Neumann subgroups, but not stated explicitly. After sending our paper we find works of Magnus( [9]), Brennen, Lyndon ( [5], [6]) and realise that our research overlap in part with those in these works. This paper is a continuation of our project to find all Cayley representations of Γ Z in P GL(2, Z) ( [10], [11], [12]) and it completes the series of works devoted to the description of Cayley's groups of P(Z). Because automorphisms groups of Γ Z is P GL(2, (Z) it gives all Cayley groups of Γ Z . To this purpose we extended the original definition of Neumann subgroup (comp. [8], [16] ) to subgroups of M . Analogously to the "modular" definition from papers of Magnus ([9]), Tretkoff ([16]), Brenner-Lyndon ( [5]) we state the following.
In the paper we get the following description of all Cayley representations of Γ Z . If S is any Cayley group of Γ Z then the following conditions are equivalent: 1. S is a Neumann subgroup of M ; 2. the set {τ n , τ n ν : n ∈ Z} form a complete system of distinct right coset representatives of S in M ; 3. there exist an involution ι : Z → Z satisfying If S ⊂ M the description of this structure is contained in the Theorem 3.1 from the Brenner-Lyndon paper [5] (see also [15]). We prove this equivalence in the more general situation of subgroups in M and in contrast to their algebraic proof our one is geometrical and uses technics involving the distant graph of P(Z). Moreover it is possible to obtain the following presentation of S: S = σ n | σ n σ ι(n) = σ n σ ι(n)+δn σ ι(n−1) = 1, n ∈ Z We do not include the proof of this fact because it is long and laborious but it consists in the typical application of the Reidemeister-Schreier procedure. We will prove in the forthcoming paper that if S is a Neumann subgroup then S is a free product of some numbers of groups of order 2, groups of order 3 and infinite cyclic groups, likewise subgroups of M . The difference is that if a Neumann subgroup is not contained in M , it has to posses free generators of negative determinant. Moreover it is possible to retrieve the set of independent generators from the above presentation. Additionally, if S ⊂ M then S = S ∩ M is a normal, nonparabolic subgroup of index 2 of S.
In the last section we describe in details structures of both groups S and S and the connection between them. Let r 2 , r 3 denote the numbers of a free generators of order 2 and 3, respectively. Then let r ± ∞ denote the number of free generators with the determinant equal to ±1, and r ∞ = r + ∞ + r − ∞ . Then we have the following restrictions Moreover the group S is a free product of 2 r 2 groups isomorphic with C 2 , 2 r 3 subgroups isomorphic with C 3 and 2 r ∞ − 1 subgroups isomorphic with Z.
The proof of this facts we postpone to next paper because we need to use the coset graph method, which is not presented it this article. Finally using the construction of an involution Z from the work [12] we show an realization of each group with the above parameters.

Cayley representations of the Z-distant graph
For the purpose of this subsection it is more convenient to use the language of matrices so we treat the extended modular groups as a quotient of GL(2, Z): M P GL(2, Z) = GL(2, Z)/{±I * }, where I * denotes the identity matrix. The elements of GL(2, Z) we will also denote by Greek lowercases with the asterisk as a superscript and their projections by the natural homomorphism Π onto P GL(2, Z) by Greek lowercases (the same as the elements of M ).
We aim to show that Neumann subgroups of M are precisely Cayley representations of the distant graph Γ Z of P(Z) Q . Since for every graph its Cayley representation can be treated as a subgroup of its automorphisms group we need the following observation. Proof. We use the notion of a maximal clique in a graph: the set of pairwise adjacent vertices is called a clique. If a clique is maximal with this property then it is called a maximal clique. We also will need a notion of a harmonic quadruple in a distant graph. For the definition we refer to [7], p.787. By Lemma 1 of ([10]) we know that a subset {v 1 By inspection of Fig. 1 one can check that for every maximal clique C and every v ∈ V (Γ Z ) there exists a finite sequence of harmonic quadruples (Q 1 , . . . , Q n ) such that Now we are in a position to start the proof. Obviously P GL(2, Z) is a subgroup of Aut(Γ Z ) and P GL(2, Z) acts transitively on the set of ordered maximal cliques of Γ Z . Let α ∈ Aut(Γ Z ) and C = (v 1 , v 2 , v 3 ) be an arbitrarily chosen ordered maximal clique in Γ Z . There exist η ∈ P GL(2, Z) that sends Fix an arbitrary vertex v ∈ V (Γ Z ) \ C and let (Q 1 , . . . , Q n ) satisfy 1., 2. and 3. Since every automorphism of Γ Z sends a harmonic quadruple to a harmonic one, every automorphism that fixes any of three members of a harmonic quadruple necessary fixes the fourth one. Therefore a simple induction argument yields (η • α)v = v. We have shown that η • α = I := Π(I * ).
Proof. Assume that S ⊂ M is a Neumann subgroup. Then, S treated as a subgroup of P GL(2, Z), by definition acts on P(Z) freely and sharply vertextransitively, and thus by the Sabidussi theorem S is a Cayley representation of Γ Z .
Conversely, for a Cayley representation S of Γ Z and each Every Neumann subgroup of M , hence a Cayley representation of Γ Z , defines some involution of Z in the following way. First observe that having a Cayley representation ( S, G, ϕ) we may assume that ϕ(1) = e. Indeed, let ( S,G,φ) be a Cayley representation of Γ Z . We get the required representation by putting From the proof of the Sabidussi theorem it follows that if in the Cayley representation of Γ Z e is labeled by 1 then ϕ(α) = αe no matter of the choice of S. Therefore from now on we will not indicate ϕ in the Cayley representations. Now the neighbourhood of e consists of vertices v n = ± n 1 ∈ P(Z), n ∈ Z, ι is an involution. The equality σ k e = v k , k ∈ Z, applied twice to n and ι(n) yields Note that δ n = det σ n = det σ ι(n) = δ ι(n) . We also have and straightforward calculations shows that σ −1 n σ n+1 = σ ι(n)−δn . Therefore the right lower term of σ * ι(n)−δn equals to −(ι(n + 1) + δ n+1 ). It follows that the involution ι satisfies In the sequel we frequently will use the equivalent form of (2.3):

4)
∈ {−1, +1}. We will need the following equality Conversely, given an involution of Z satisfying (2.3) we can define a subgroup of P GL(2, Z) generated by elements defined by (2.1). We will say that such subgroups of M are generated by an involution. Note that (2.4) yields the following relations in the subgroups generated by involutions: (2.7) It is possible, but a bit laborious, to show that in fact those relations form presentations of such groups. We will need versions of those relations in GL(2, Z): (2.9) Theorem 2.3. A subgroup of M is a Neumann subgroup iff it is generated by an involution.
Proof. It was already shown that every Neumann subgroup of M is always generated by an involution.

Structure of Neumann subgroups of the extended modular group
In this section we provide two theorems that completely describe structure of Neumann subgroups of the extended modular group. The proof of the first one can be found in either [5] or derived from [15]. The second theorem is new but we postpone its proof to the forthcoming paper since it requires methods we do not develop in this paper. The following theorem describes the structure of Neumann subgroups of the modular group. From the Kurosh Subgroup Theorem a subgroup S of the modular group is a free product of r 2 subgroups of order 2, r 3 subgroups of order 3 and r ∞ of infinite cyclic subgroups. The fact that any group H is such a free product we express saying that H has (r 2 , r 3 , r ∞ )-structure.
Theorem 3.1 ( [5], [15]). If S is a Neumann subgroup of M then S has (r 2 , r 3 , r ∞ )-structure subject to the conditions that r 2 + r 3 + r ∞ = ∞ and if r ∞ is finite then its even. Moreover every structure satisfying the above conditions is realized by some Neumann subgroup of M .
If we assume that a Neumann subgroup of M is not contained in M then the situation becomes essentially more complicated and we describe it in the theorem below. As was announced in Introduction we postpone the proof to the forthcoming paper. We only make here some remarks on a number of independent generators in groups S and S = S ∩ M . It is easy to observe that the set where α ∈ S is an arbitrary element of the determinant equal to −1, generates S. Assume now that α is taken to be equal to some of σ n 's and the set D of generators of S with the negative determinant is finite. Then the set αD ∪ Dα −1 \ {I} has odd cardinality. Moreover all other generators are doubling. We are not able to prove the below theorem by this method since it does not allow to prove that there is a subset of {σ n } of independent generators (which actually is true). This is just a hint regarding statements about cardinality of the sets of different kinds of generators.
Recall that r 2 , r 3 denote the number of independent generators of order 2 and 3, respectively, and that r ± ∞ denote the number of free generators with the determinant equal to ±1, r ∞ = r + ∞ + r − ∞ . Theorem 3.2. Let S be a Neumann subgroup of M that is not entirely contained in M and let S = S ∩ M . Then S S, S/S C 2 and S is never a semi-direct product of S and a subgroup isomorphic to C 2 . Moreover • S has ( r 2 , r 3 , r ∞ )-structure subject to the conditions that 1. r 2 + r 3 + r ∞ = ∞, 2. r − ∞ ≥ 1 and if r + ∞ is finite then its even, 3. r 2 + r 3 + r + ∞ 2 ≥ r − ∞ and all independent generators of finite order are elliptic; • if S has (r 2 , r 3 , r ∞ )-structure then the following equalities hold (r 2 , r 3 , r ∞ ) = (2 r 2 , 2 r 3 , 2 r ∞ − 1).
Analogously to the situation in Theorem 3.1 it is possible to realize any admissible structure for a Neumann subgroup of the extended modular group. Recall that in order to describe some Neumann subgroup it is enough to define appropriate involution ι of the set of integers. The recursive method of such a construction is given in [12]. For the sickness of completeness we describe this method below. A mapι : {k, . . . , k + l} −→ {k, . . . , k + l}, k ∈ Z, l ≥ 0, is called a building involution if: •ι is an involution of {k, . . . , k + l}; •ι(k) = k + l; •ι satisfies (2.3) for each n = k, . . . , k + l − 1.
If we assume that δ kn = 1 for all n then it follows immediately from the construction that such defined ι satisfies (2.3).