© Hindawi Publishing Corp. POWERS OF A PRODUCT OF COMMUTATORS AS PRODUCTS OF SQUARES

We prove that for any odd integer N and any integer n>0, the Nth power of a product of n commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of 2n

1. Introduction.Lyndon et al. [2] have shown that the product of n commutators in a nonabelian free group can be written as a product of 2n + 1 squares of elements and there are commutators for which the number 2n + 1 of squares is the minimum number such that the product of these commutators can be written as a product of squares.Recently, Akhavan-Malayeri [1] proved, for an odd integer n, that [x, y] n of two distinct elements of a free generating set of a nonabelian free group is not a product of two squares but it is the product of three squares.We generalize these results in the following theorem.
Theorem 1.1.Let F be a free group with a basis of distinct elements x 1 ,...,x 2n , and N any odd integer.Then there exist elements u 1 ,...,u m in F such that if and only if m ≥ 2n + 1.
Note that the theorem for even N is not true since the element in the left-hand side of the above equation is actually a square.The proof of this theorem is almost mutatis mutandis as the proof of the main result of [2].Throughout this note, [x, y] = x −1 y −1 xy and [x,y,z] = [[x, y], z] for all elements x, y, z of a group G, and G denotes the derived subgroup of G.

Proof of the main result
Proof of Theorem 1.1.We show first that this equation has a solution for m = 2n + 1, hence trivially for m ≥ 2n + 1.Since N is odd, there is an integer k such that N = 2k + 1.Thus it is enough to show that, for any element v of F , we can express the element as a product of 2n + 1 squares.We argue by induction on n.If n = 1, then by the following well-known identity this case is proved: (2.1) Assume n > 1 and suppose inductively that for some elements u 1 ,...,u 2n−1 in F .Now by the identity (2.1) we can write for some elements U , V , and W in F , and so which completes the induction.This first part of the proof is essentially well known in a topological context: the nonorientable surface formed by attaching one cross-cap and n handles to a sphere (the connected sum of 1 projective plane and n tori) is homeomorphic to the surface obtained by attaching 2n + 1 cross-caps (the connected sum of 2n + 1 projective planes).In this context, the identity (2.1) is just the handle calculus that says cross-cap + handle = 3 cross-caps.
For the converse, we suppose that the equation holds.Let G be the group with the following presentation: The equation would also hold in G since G is a quotient of F .So we have for some elements v 1 ,...,v m in G. Since N is odd, N = 2t + 1 for some integer t.Since G is nilpotent of class 2 and y 2 i = 1 for each i, we have [y i ,y j ] 2 = 1 and all the commutators are in the center of G, so the latter equation can be rewritten as where a ik ∈ Z 2 and z k ∈ G for all i ∈ {1,...,2n} and all k ∈ {1,...,m}.Since z 2 k = 1 for all k, we have (2.10) If A is the matrix A = (a ij ) over Z 2 , and A i = (a i1 ,...,a im ) is the ith row of A, then from the relation we conclude that, taking inner products, (2.11) We conclude that A•A T = B, where A T is the transpose of A, and B is the direct sum of n matrices of the form 1 0 0 1 , and hence has rank 2n.It follows that rank(A) ≥ 2n.But the equation

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.