ON A PROBLEM ON EXPLICIT EMBEDDINGS OF THE GROUP Q

Answering a question of de la Harpe and Bridson in the Kourovka Notebook, we build the explicit embeddings of the additive group of rational numbers ℚ in a finitely generated group G. The group G in fact is two-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.


Introduction
In 1999, de la Harpe and Bridson grouped a few "well-known" questions as Problem 14.10 in Kourovka notebook [7].These questions mainly concern explicit embeddings of some (finitely or infinitely presented) countable groups into finitely generated groups.In particular, they posed the following possibly easier problem.Problem 1.1 [7,from Problem 14.10].Find an explicit embedding of the additive group of rational numbers Q in a finitely generated group; such a group exists by [4,Theorem IV].
An explicit embedding mentioned in this problem is possible and, moreover, the finitely generated group G can in fact be two-generator.This, of course, stresses the similarity of the constructed embedding with the embedding of [4] because the latter, too, is an embedding into a two-generator group.
In the current paper, we suggest two mechanisms to build the embedding mentioned.The first argument uses the operation of wreath products, while the second argument is based on Higman-Neumann-Neumann extensions (HNN extensions) of groups.
For general group-theoretical background information, we refer to [5,15].More specific information on wreath products can be found in [12], and on generalized free products and HNN extensions can be found in [6,14].For original methods and techniques of work with embeddings of countable groups in two-generator groups, we refer to wellknown articles [2,3,4,11].

An embedding based on wreath products
In our first construction, the group G will be built as a two-generator subgroup in the nested cartesian wreath product W = (Q Wr C)WrZ, where C = c and Z = z are infinite cyclic groups (written multiplicatively).Firstly, for each positive integer n, choose in the base subgroup Q C of the cartesian wreath product Q Wr C the elements ϕ n and ψ n : (2.1) The reason of such selection is in the following relations: 2) The first of the relations (2.2) follows from (2. 3) The second of the relations (2.2) trivially follows from the fact that Q C is abelian.
In the base subgroup (Q Wr C) Z of W, take an element α defined as (2.4) Put G = α,z and define the embedding That Φ is a homomorphism and an injection could be checked directly.But to avoid very long calculations, we consider the structure of the commutator [α z −n ,α] first: (the last two lines follow from (2.2)).This means that [α z −n ,α] is nothing else but the image ϕ * n of the coordinate element ϕ n in the "the first copy" of the group Q Wr C in W: (2.7)

Vahagn H. Mikaelian 2121
Therefore the elements do form a subgroup isomorphic to Q in G, and the mapping Φ is injective.Finally, it easily follows from the equalities that Φ is a homomorphism (where in the formula above "+" is used in the sense that all the coordinates of the summands are added as rational numbers).

An embedding based on HNN extension
This time, the group G will be built as a certain HNN extension of the free product H = Q * F 2 , where F 2 = x, y is a free group of rank 2. For each positive integer n in the group H, choose a pair of elements Evidently, for any n > 0, the words u n and v n are reduced in H and, in particular, are nontrivial.Therefore, the map f : u n → v n , n > 0 can be continued to an isomorphism f of subgroups of H. Take an "external" stable letter t and build the HNN extension N = H,t defined by Put G = x,t ≤ N and define the embedding Here, too, it can be checked directly that this is a homomorphism and is an injection.But, again, to avoid long calculations, we consider the above components of Ψ(m/n).
holds.On the other hand, The product of (3.5) and (3.6) is then equal to 1/n (in fact, to the copy of 1/n inside the HNN extension of the free product Q * F 2 ).Therefore the elements do form a subgroup isomorphic to Q.The map Ψ is injective, and evidently is a homomorphism.We may also notice that, in fact, G = N because N contains a set generating G, namely,

Some comparison and additional properties for embeddings
Of these two embeddings Φ and Ψ, the second is much closer to the original embedding of Higman et al. [4].However, the embedding Φ is much more economical, because the abelian group Q is embedded into a soluble group of length 3. Whereas Ψ embeds Q in a group that contains a subgroup isomorphic to the free group of infinite rank (and, therefore, generates the variety of all groups).On the other hand, for the embedding Φ, we needed the fact that Q is abelian; it was required for the equality [ψ n ,ψ k ] = 1 in (2.2).This means that the concept of Φ can only be used to build explicit embedding of countable abelian groups given by their generators.Whereas the concept of Ψ is suitable for nonabelian groups, as well.
However, there are a few more complex versions of the embedding Φ that allow to also build explicit embeddings of nonabelian groups.But here we restrict our consideration to this simpler form of Φ and refer to [10] for further variations of the argument.
Another addition to the above construction of Φ is that the group G can be modified to have a full order (in the sense of [13]) that continues the "normal" order in Q.That is, G can be built to have an order relation "<" such that (i) "<" is a linear order on G; (iii) for any g,h,t ∈ G, if g < h, then also gt < ht and tg < th.
For further development of this subject, see our recent work [8,9,10].The embeddings Φ and Ψ both preserve the torsion freeness of Q.The two-generator group built for Φ is torsion free because it is a subgroup of the torsion-free group W [12].
That the two-generator group built for Ψ is torsion free follows from the following.Since Q and F 2 both are torsion free, the group H = Q * F 2 evidently is torsion free.By [6,Theorem 2.4], if the group G = N has an element g of finite order, then it is conjugate with an element of the base subgroup H. Since the latter is torsion free, g = 1 holds.
Finally, we notice that the embedding Φ is subnormal: Z , and this base subgroup is normal in W. Subnormality of embeddings of countable groups into two-generator groups was first proved by Dark in [1].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: