A Remark on the Intersection of the Conjugates of the Base of Quasi-hnn Groups

Quasi-HNN groups can be characterized as a generalization of HNN groups. In this paper, we show that if G * is a quasi-HNN group of base G, then either any two conjugates of G are identical or their intersection is contained in a conjugate of an associated subgroup of G.


Introduction.
In [8, Lemma 3.15, page 152], Scott and Wall proved that if G = G 1 * C G 2 is a nontrivial free product with amalgamation group, then either gG 1 g −1 ∩G i is a subgroup of a conjugate of C, or i = 1 and g ∈ G 1 , so that gG 1 g −1 ∩G i = G 1 .In this paper we generalize such a result to groups acting on trees with inversions and then apply the result we obtain to a new class of groups called quasi-HNN groups, introduced in [2].This paper is divided into five sections.In Section 2, we give basic definitions.In Section 3, we have notations related to groups acting on trees with inversions.In Section 4, we discuss the intersections of vertex stabilizers of groups acting on trees with inversions.In Section 5, we apply the results of Section 4 to a tree product of groups and of quasi-HNN groups.

Groups acting on graphs.
In this section, we begin by recalling some definitions taken from [3,7].First we give formal definitions related to groups acting on graphs with inversions.By a graph X we understand a pair of disjoint sets V (X) called the set of vertices and E(X) called the set of edges, with V (X) nonempty, equipped with two maps E(X) → V (X) × V (X), y → (o(y), t(y)), and E(X) → E(X), y → y, satisfying the conditions y = y and o(y) = t(y) for all y ∈ E(X).The case y = y is possible for some y ∈ E(X).For y ∈ E(X), o(y) and t(y) are called the ends of y and y is called the inverse of y.There are obvious definitions of trees, morphisms of graphs, and Aut(X), the set of all automorphisms of the graph X which is a group under the composition of morphisms.We say that a group G acts on a graph X if there is a group homomorphism φ : G → Aut(X).If x ∈ X (vertex or edge) and g ∈ G, we write g(x) for (φ(g))(x).Thus if g ∈ G and y ∈ E(X), then g(o(y)) = o(g(y)), g(t(y)) = t(g(y)), and g(y) = g(y).The case g(y) = y for some g ∈ G and y ∈ E(X) may occur.That is, G acts with inversions on X.
We have the following definitions related to the action of the group G on the graph X.
(1) If x ∈ X (vertex or edge), define G(x) to be the set G(x) = {g(x) : g ∈ G}.This set is called the orbit that contains x.
(2) If x, y ∈ X, define G(x, y) to be the set G(x, y) = {g ∈ G : g(x) = y}, and G(x, x) = G x , the stabilizer of x.Thus, G(x, y) ≠ ∅ if and only if x and y are in the same orbit.If y ∈ E(X) and u ∈ {o(y), t(y)}, then it is clear that G y = G y and G y ≤ G u .(3) If X is connected, then a subtree T of X is called a tree of representatives for the action of the group G on X if T contains exactly one vertex from each vertex orbit, and the subgraph Y of X containing T is called a fundamental domain if each edge of Y has at least one end in T , and Y contains exactly one edge y from each edge orbit such that G(y, y) = ∅, and exactly one pair x, x from each edge orbit such that G(x, x) ≠ ∅.

Notations.
Let G be a group acting on a tree X with inversions, let T be a tree of representatives for the action of G on X, and let Y be a fundamental domain.We have the following notations.
(1) For any vertex v of X, let v * be the unique vertex of That is, v and v * are in the same vertex orbit.(2) For each edge y of Y , define the following:

On the intersection of vertex stabilizers of groups acting on trees with inversions.
In this section, G will be a group acting on a tree X with inversions, T is a tree of representatives for the action of G on X, and Y is a fundamental domain.We have the following definition.

Definition 4.1. A word w of G means an expression of the form w
The following concepts are related to the word w defined above: (i) n is called the length of w and is denoted by |w| = n, (ii) w is called a trivial word of G if |w| = 0 (or w = g 0 ), (iii) the value of w, denoted by [w], is defined to be the element of G: (iv) the inverse of w, denoted by w −1 , is defined to be the word of G: Lemma 4.2.Let w be a nontrivial reduced word of G and let a ∈ G o(w) be such that Then w 0 is a nontrivial closed word of G such that [w 0 ] = 1, the identity element of G.
Therefore by [4, Corollary 1], w 0 is not reduced.Since w is reduced, then w −1 is reduced.Therefore the only possibility that makes w 0 not reduced is ..,n.By taking x i = L i (+y i ), we see that a ∈ G x i for i = 1,...,n.By the corollary of [5, Theorem 1], x 1 ,...,x n is a reduced path in X from o(w) to [w](t(w)).This completes the proof.

Theorem 4.3. For any two vertices u and v of X, G
where x is an edge in the reduced path in X joining u and v.
Then it is clear that u ≠ v.We need to show that h is in G x , where x is an edge in the reduced path in X joining u and v.We have u = f (u * ) and v = g(v * ), where f and g are in G and u * and v * are the unique vertices of This contradicts the assumption that u ≠ v.By Lemma 4.2, there exists a reduced path p 1 ,...,p n in X joining o(w) = u * and [w](t(w)) = f −1 g(v * ) such that a ∈ G p i for i = 1,...,n.Let x i = f (p i ), i = 1,...,n.Then it is clear that x 1 ,...,x n is the reduced path in X joining u and v and h ∈ G x i for i = 1,...,n.This implies that G u ∩ G v ≤ G x i for i = 1,...,n.This completes the proof.
We have the following corollaries of Theorem 4.3.

Corollary 4.4. For any edge
Corollary 4.5.Let u and v be two vertices of X and let x 1 ,...,x n be the reduced path in X joining u and v such that Corollary 4.6.Let u and v be two vertices of X such that G u ≠ G v and let x be an edge in the reduced path in X joining u and v.
Corollary 4.7.Let u be a vertex of X and let v be a vertex of T .Then G u ∩G v ≤ G x , where x is an edge in the reduced path in X joining u and v, or

Applications.
In this section Theorem 4.3 and its corollaries are applied to a nontrivial tree product of groups introduced in [1] and of quasi-HNN groups introduced in [2].
In [5,Lemma 8], Mahmood showed that if G = * i∈I (A i ,U jk = U kj ) is a nontrivial tree product of the groups A i , i ∈ I, then there exists a tree X on which G acts without inversions such that any tree of representatives for the action of G on X equals the fundamental domain and for every vertex u of X and every edge x of X, G u is a conjugate of A i for some i in I and G x is a conjugate of U ik for some i, k in I.
In [6, Lemma 5.1], Mahmood and Khanfar showed that if G * is the quasi-HNN group G * = G, t i ,t j | rel G, t i A i t −1 i = B i , t j C j t −1 j = C j , t 2 j = c j , i ∈ I, j ∈ J , then there exists a tree X on which G * acts with inversions such that G * is transitive on V (X) and for every vertex v of X and every edge Then by Theorem 4.3, the following two propositions hold.
Proposition 5.1.Let G = * i∈I (A i ,U jk = U kj ) be a nontrivial tree product of the groups A i , i ∈ J. Then for any g in G and i and s in I, either gA i g −1 ∩ A s is contained in a conjugate of U jk or i = j, g ∈ A i , and (5.1)

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: where x is an edge in the reduced path in X joining u and u * , or u * = u and G u ∩ G u * = G u .Corollary 4.9.For any edge y of Y , G (o(y)) * = G (t(y)) * , or G (o(y)) * ∩G (t(y)) * ≤ G m , where m is an edge in the reduced path in T joining (o(y)) * and (t(y)) * .