© Hindawi Publishing Corp. ON CENTRALIZERS OF ELEMENTS OF GROUPS ACTING ON TREES WITH INVERSIONS

A subgroup H of a group G is called malnormal in G if it satisfies the condition that if g ∈ G and h ∈ H , h ≠ 1 
such that g h g − 1 ∈ H , then g ∈ H . In this paper, we show 
that if G is a group acting on a tree X with inversions such 
that each edge stabilizer is malnormal in G , then the 
centralizer C ( g ) of each nontrivial element g of G is in a 
vertex stabilizer if g is in that vertex stabilizer. If g is 
not in any vertex stabilizer, then C ( g ) is an infinite cyclic 
if g does not transfer an edge of X to its inverse. Otherwise, 
 C ( g ) is a finite cyclic of order 2.

By a graph X, we understand a pair of disjoint sets V (X) and E(X) with V (X) nonempty together with a mapping E(X) → V (X) × V (X), y → (o(y), t(y)), and a mapping E(X) → E(X), y → ȳ, satisfying the conditions that ȳ = y and o( ȳ) = t(y), for all y ∈ E(X).The case ȳ = y is possible for some y ∈ E(X).For y ∈ E(X), o(y) and t(y) are called the ends of y, and ȳ is called the inverse of y.There are obvious definitions of subgraphs, 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 of graphs.For more details, see [4,5].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).If y ∈ E(X) and g ∈ G, then g(o(y)) = o(g(y)), g(t(y)) = t(g(y)), and g( ȳ) = g(y).The case g(y) = ȳ for some g ∈ G and some y ∈ E(X) may occur.That is, G acts with inversions on X.
We have the following notations related to the action of the group G on the graph X.
(1) If x ∈ X (vertex or edge), we define G(x) = {g(x) : g ∈ G}, and this set is called the orbit containing x. (2) If x, y ∈ X, we define 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 G orbit.It is clear that if v ∈ V (X), y ∈ E(X), and u ∈ {o(y), t(y)}, then G(v, y) = ∅, G ȳ = G y , and G y ≤ G u .

3.
The structure of groups acting on trees with inversions.In this section, we summarize the structure of groups acting on trees with inversions obtained by [4].Definition 3.1.Let G be a group acting on a tree X, and T and Y two subtrees of X such that T ⊆ Y .Then, T is called a tree of representatives for the action of G on X if T contains exactly one vertex from each G vertex orbit, and Y is called a fundamental domain for the action G on X, if each edge of Y has at least one end in T , and Y contains exactly one edge y (say) from each G edge orbit such that G( ȳ,y) = ∅, and exactly one pair x and x from each G-edge orbit such that G( x, x) ≠ ∅.It is clear that the properties of T and Y imply that if u and v are two vertices of T such that G(u, v) ≠ ∅, and if x and y are two edges of Y such that G(x, y) ≠ ∅, then u = v and x = y or x = ȳ.
Let T and Y be as above.Define the following subsets Y 0 , Y 1 , and Y 2 of edges of Y as follows: (1) For the rest of this section, G will be a group acting on a tree X with inversions, T will be a tree of representatives for the action of the group G on X, and Y will be a fundamental domain for the action of G on X such that T ⊆ Y .We have the following definitions.Definition 3.2.For each vertex v of X, define v * to be the unique vertex of T such that G(v, v * ) ≠ ∅.That is, v and v * are in the same G vertex orbit.It is clear that if v is a vertex of T , then v * = v and, in general, for any two vertices u and v of X such that G(u, v) ≠ ∅, we have u * = v * , and G u and G v are conjugate by an element of G.That is, for every element b of G u , there exist g of G and a of G v such that b = gag −1 .Definition 3.3.For each edge y of Y 0 ∪Y 1 ∪Y 2 , define [y] to be an element of G(t(y), (t(y)) * ).That is, [y] satisfies the condition that [y]((t(y )) * )= t(y), and to be chosen as follows: Define [ ȳ] to be the element From above we see that A group is termed a quasifree group if it is a free product of copies of C ∞ and C 2 , where C ∞ denotes infinite cyclic group and C 2 a cyclic group of order 2.
The following are examples of quasifree groups: (1) every free group is a quasifree group.That is, a free product of copies of C ∞ and a zero number of copies of C 2 ; (2) the group of the presentation x, y, z Lemma 3.4.Let G, X, Y , and T be as above such that the stabilizer of each vertex of X is trivial.Then, G is a quasifree group.

Proof. By [4, Theorem 3.6], G has the presentation
Since the stabilizer of each vertex of X is trivial, G v , G m , G y , and Then, G is a free product of C ∞ generated by y, and C 2 generated by x.
This implies that G is a quasifree group.This completes the proof.

Corollary 3.5. Let G, X, Y , and T be as above, and
Then H is a quasifree group.Definition 3.6.For each edge y of Y , define the following.
(3) Define φ y to be the map φ y : It is clear that φ y is an isomorphism.(4) Define δ y to be the element Consequently, δ y ∈ G y , δ ȳ = δ y , and φ y (δ y ) = δ y .Definition 3.7.By a word w of G, we mean an expression of the form Let w be the word defined above.We have the following concepts: (a) w is called reduced if w contains no expression of the form

The main result.
The main result of this section is the following theorem.Theorem 4.1.Let G be a group acting on a tree X with inversions such that each edge stabilizer is malnormal in G. Let g be a nontrivial element of G and C(g) the centralizer of g in G.Then, (i) if g is in a vertex stabilizer, then C(g) is in that vertex stabilizer; (ii) if g is not in any vertex stabilizer, then C(g) is an infinite cyclic subgroup of G if g does not transfer an edge of X to its inverse.Otherwise, C(g) is a finite cyclic subgroup of G of order 2.
Proof.(i) Let v be a vertex of X such that g ∈ G v .We need to show that C(g) is contained in G v .Let f be an element of G such that f g = gf .We need to show that f is in G v .We consider two cases.
Case 1 (g is in G x , where x is an edge of X such that v is an end of x).Since the edge stabilizer for each edge of Case 2 (g is not in any edge stabilizer of G).Then, there exists a unique vertex v * of T such that G(v, v * ) ≠ ∅, that is, v and v * are in the same G vertex orbit.Then, there exist a ∈ G and b The only way that the indicated word can fail to be reduced is that g n hg −1  n ∈ G −yn .Replacing a subword of the form y i • g i • ȳi if g i ∈ G −y i by φ y i (g i δ y i ), or replacing a subword of the form y i • g i • y i if g i ∈ G y i and G(y i , ȳi ) ≠ ∅, by φ y i (g i δ y i ), we see that each (ii) Now, suppose that g is not in any vertex stabilizer of G.Then, C(g) has trivial intersection with each vertex stabilizer of G.If a ≠ 1 is in C(g), and a is in a vertex stabilizer G v of G, for the vertex v of X, then g is in C(a) and, by above, g is in G v .This contradicts the assumption that g is not in any vertex stabilizer of G. Hence, by Corollary 3.5, C(g) is a free product of a number of infinite cyclic groups and a number of finite cyclic groups of order 2. Since C(g) has a nontrivial center, and the center (see [3, Corollary 4.5, page 211]) of free product of groups of more than one factor is trivial, then C(g) is an infinite cyclic groups, or C(g) is a finite cyclic groups of order 2. If g transfers an edge of X to its inverse, then, by [7,Corollary 4.3], C(g) is a finite cyclic group of order 2. Otherwise, C(g) is an infinite cyclic group.This completes the proof.Definition 4.2.Let n be a positive integer and g a nontrivial element of the group H.We say that g has at most nth root if whenever g = a n = b n , for a, b in H, then a = b.
In the next corollaries, the group G satisfies the hypothesis of Theorem 4.1.

Corollary 4.3. Any element of G that is not in any vertex stabilizer of G has at most nth root.
Proof.Let g, a, and b be elements of G such that g is not in any vertex stabilizer of G, and g = a n = b n .We need to show that a = b.By Theorem 4.1, C(g) is an infinite cyclic group or is a finite cyclic group of order 2.
Since ga = ag and gb = bg, then a and b are in C(g).Then, it is clear that g = a n = b n implies that a = b.This completes the proof.Corollary 4.4.Let g be an element of G.Then, g is not in any vertex stabilizer of G if and only if g n is not in any vertex stabilizer of G, where n is a positive integer.
Proof.Since g n commutes with g, then g n is in C(g) which, by Theorem 4.1, is not in any vertex stabilizer of G and the result follows.This completes the proof.Corollary 4.5.Let f and g be two elements of G, and m and n two positive integers such that f and g are not in any vertex stabilizer of G and f m g n = g n f m .Then f g = gf .
Proof.From Corollary 4.4, we get

5.
Applications.This section is an application of Theorem 4.1 and its corollaries.Free groups, free product of groups, free product of groups with amalgamation subgroup, tree product of groups, and HNN groups are examples of groups acting on trees without inversions.A new class of groups called quasi-HNN groups, defined in [2], are examples of groups acting on trees with inversions.In fact, free product of groups, free product of groups with amalgamation subgroup are special cases of tree product of groups and free groups and HNN groups are special cases of quasi-HNN groups.
Proposition 5.1.Let G = * i∈I (A i ,U jk = U kj ) be a nontrivial tree product of the groups A i , i ∈ I, such that U ij are malnormal subgroups of G. Let g be a nontrivial element of G and C(g) be the centralizer of g in G.Then, , for all i ∈ I, then C(g) is an infinite cyclic group and g has at most nth root.
Proof.By [6], 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 corresponding fundamental domain for the action of G on X, and for every vertex u of X and every edge x of X, G u is isomorphic to A i , i ∈ I, and G x is isomorphic to U ik for some i, k in I.Moreover, G contains no invertor elements.Therefore, by Theorem 4.1 and Corollary 4.3, the proof of Proposition 5.1 follows.This completes the proof.
such that A i , B i , and C j , i ∈ I and j ∈ J, are malnormal subgroups of G * .Let g a nontrivial element of G * and C(g) the centralizer of g in G * .Then, (i) C(g) is in a conjugate of G if g is in a conjugate of G; (ii) if g is not in a conjugate of G, then C(g) is an infinite cyclic group or a finite cyclic group of order 2 and g has at most nth root.
Proof.By [8, Lemma 5.1], 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 x of X, G * v is isomorphic to G and G * x is isomorphic to A i , i ∈ I, or isomorphic to C j , j ∈ J.Moreover, G * contains the invertor elements conjugate to an element t j , j ∈ J. Therefore, by Theorem 4.1, the proof of Proposition 5.4 follows.This completes the proof.
Corollary 5.5.Let G * be the HNN group such that A i and B i , i ∈ I, are malnormal subgroups of G * .Let g a nontrivial element of G * and C(g) the centralizer of g in G * .Then, (i) C(g) is in a conjugate of G if g is in a conjugate of G; (ii) if g is not in a conjugate of G, then C(g) is an infinite cyclic group and g has at most nth root.
Corollary 5.6.If g is a nontrivial element of a free group F , then the centralizer C(g) of g in F is an infinite cyclic group and g has at most nth root.
w and is denoted by |w| = n; (e) the inverse of w denoted by w −1 is defined as the word w

Proposition 3 . 8 .
Every element of G is the value of a closed and reduced word of G.Moreover, if w is a nontrivial closed and reduced word of G, then [w] is not the identity element of G.Moreover, if w 1 and w 2 are two closed and reduced words of G such that w 1 and w 2 are of the same type and of the same value, then |w 1 | = |w 2 |.Proof.See [5, Corollary 3.6].
and of value h.That is, (o(y)) * = (t(y)) * = v * and [w] = h.Since h ∈ G −yn , then w • h and h • w are reduced words of G of value 1, the identity element of G. Therefore, by Proposition 3.8, the word w

(4. 1 )Corollary 4 . 6 .
This completes the proof.Let f and g be two elements of G such that f and g are not in any vertex stabilizer of G and f ∈ C(g).Then C(f ) = C(g).Proof.By Theorem 4.1, C(f ) and C(g) are cyclic subgroups of G.Then, there exist two elements a and b of G such that a and b are not in any vertex stabilizer of G, C(f ) = a , and C(g) = b .It is clear that if a ∈ C(g), then C(f ) = C(g).There exist two positive integers m and n such that f = a m and g = b n .Since f g = gf , a m b n = b n a m .Then, Corollary 4.5 implies that ab = ba.This implies that ab n = b n a. Then a ∈ C(b n ) = C(g).This completes the proof.
be a free product of the groups A and B with amalgamation subgroup C such that C is a malnormal subgroup of G. Let g be a nontrivial element of G and C(g) the centralizer of g in G.Then, Let g be a nontrivial element of G and C(g) the centralizer of g in G.Then, C B (i) C(g) is in a conjugate of A or B if g is in a conjugate of A or B; (ii) if g is not in a conjugate of A or B, then C(g)is an infinite cyclic group and g has at most nth root.Corollary 5.3.Let G = A * B be a free product of the groups A and B. (i) C(g) is in a conjugate of A or B if g is in a conjugate of A or is in a conjugate of B; (ii) if g is not in a conjugate of A or B, then C(g) is an infinite cyclic group and g has at most nth root.Proposition 5.4.Let G * be the quasi-HNN group