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

Abstract

There is presented an infinite class of subgroups of the modular group 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 PSL(2, Z) isomorphic with Z2, Z3 and Z subject to the restriction that the number of copies of Z is 0 or 2. The proof technique is based on a 1-1 correspondence between some involutions ι of Z that fulfill the equation ι(ι(n) − δn) = ι(n + 1) + δn+1, δn = ± 1, δ ι(n) = δn, and groups from this class.

The Cayley Graph of Neumann Subgroups

“…After sending our paper we found works of Magnus [9], Brenner, 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 .…”

“…In [11] it was proved that the distant graph Γ Z is a Cayley one and then in [12] there were constructed uncountably many of its Cayley representation. Inherently, it was proven that its Cayley representations in the modular group are Neumann subgroups, but not stated explicitly.…”

“…The proof of this facts we postpone to the 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.…”

“…Recall that in order to describe some Neumann subgroup it is enough to define an appropriate involution ι of the set of integers. The recursive method of such a construction is given in [12]. For the sake of completeness we describe this method below.…”

The Cayley Graph of Neumann Subgroups

“…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.…”