On centralizers of elements of groups acting on trees with inversions

(1)

PII. S0161171203205305 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON CENTRALIZERS OF ELEMENTS OF GROUPS ACTING

ON TREES WITH INVERSIONS

R. M. S. MAHMOOD

Received 2 May 2002

A subgroupHof a groupGis called malnormal inGif it satisfies the condition that ifg∈Gandh∈H,h≠1 such thatghg−1_∈_{H, then}_g_∈_{H. In this paper, we} show that ifGis a group acting on a treeXwith inversions such that each edge stabilizer is malnormal inG, then the centralizerC(g)of each nontrivial element gofGis in a vertex stabilizer ifgis in that vertex stabilizer. Ifgis not in any vertex stabilizer, thenC(g)is an infinite cyclic ifgdoes not transfer an edge ofX to its inverse. Otherwise,C(g)is a finite cyclic of order 2.

2000 Mathematics Subject Classification: 20F65, 20E07, 20E08.

1. Introduction. There are many groups with the property that centralizers of nontrivial elements are cyclic. For example, the centralizers of the nontrivial elements (see [3, Problem 7, page 42]) of free groups are infinite cyclic. It has been shown in [3, Problem 28, page 196] that the centralizers of the nontrivial elements of the free product of groups are in a conjugate of a factor or infinite cyclic. In [1, Theorem 1], Karrass and Solitar proved that the centralizers of nontrivial elements of the free product of two groups with a malnormal amal-gamated subgroup are in a conjugate of a factor or infinite cyclic. In this paper, we generalize such results to groups acting on trees with inversions as follows. IfGis a group acting on a treeXwith inversions such that the stabilizerGxof

every edgexofXis a malnormal subgroup inG, then the centralizersC(g)of every nontrivial elementgofGis in a vertex stabilizerGv of a vertexvofXif

gis inGv. Ifgis not in any vertex stabilizer ofG, thenC(g)is an infinite cyclic

subgroup ofGifgdoes not transfer any edge ofXto its inverse. Otherwise, C(g)is a finite cyclic subgroup ofGof order 2.

This paper is divided into five sections. InSection 2, we introduce the con-cepts of graphs and the actions of groups on graphs. InSection 3, we have a summary of the structure of groups acting on trees with inversions and ele-mentary results. InSection 4, we discuss the structure of the centralizers of the elements of groups acting on trees with inversions.Section 5is an application of the results inSection 4.

(2)

By a graphX, we understand a pair of disjoint setsV (X)andE(X)withV (X) nonempty together with a mappingE(X)→V (X)×V (X), y→(o(y),t(y)), and a mapping E(X)→E(X), y →y, satisfying the conditions that ¯¯ 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 ofy. There are obvious definitions of subgraphs, trees, mor-phisms of graphs, and Aut(X), the set of all automormor-phisms of the graphX which is a group under the composition of morphisms of graphs. For more details, see [4,5]. We say that a groupGacts on a graphXif there is a group homomorphism φ: G→ Aut(X). If x ∈X (vertex or edge) and g ∈G, we writeg(x)for(φ(g))(x). Ify∈E(X)and g∈G, theng(o(y))=o(g(y)), g(t(y))=t(g(y)), andg(y)¯ =g(y). The caseg(y)=y¯for someg∈Gand somey∈E(X)may occur. That is,Gacts with inversions onX.

We have the following notations related to the action of the groupGon the graphX.

(1) Ifx∈X(vertex or edge), we defineG(x)= {g(x):g∈G}, and this set is called the orbit containingx.

(2) Ifx,y∈X, we defineG(x,y)= {g∈G:g(x)=y}andG(x,x)=Gx,

the stabilizer ofx. Thus,G(x,y)≠∅if and only ifxandyare in the sameGorbit. It is clear that ifv∈V (X),y∈E(X), andu∈ {o(y),t(y)}, thenG(v,y)= ∅,Gy¯=Gy, andGy≤Gu.

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 ofXsuch thatT⊆Y. Then,T is called a tree of representatives for the action ofGonXifTcontains exactly one vertex from eachGvertex orbit, andY is called a fundamental domain for the actionGonX, if each edge ofY has at least one end inT, andY contains exactly one edgey(say) from each Gedge orbit such thatG(y,y)¯ = ∅, and exactly one pairxand ¯xfrom each G-edge orbit such thatG(x,x)¯ ≠∅. It is clear that the properties ofT andY imply that ifuandvare two vertices ofTsuch thatG(u,v)≠∅, and ifxand yare two edges ofY such thatG(x,y)≠∅, thenu=vandx=yorx=y.¯

LetTandYbe as above. Define the following subsetsY0,Y1, andY2of edges

ofY as follows:

(1) Y0=E(T ), the set of edges ofT,

(2) Y1= {y∈E(Y ):o(y)∈V (T ), t(y)∈V (T ), G(¯y,y)= ∅},

(3) Y2= {x∈E(Y ):o(x)∈V (T ), t(x)∈V (T ), G(¯x,x)≠∅}.

It is clear thatGacts with inversions onXif and only ifY2≠∅.

(3)

ON CENTRALIZERS OF ELEMENTS OF GROUPS 1243

andY will be a fundamental domain for the action ofGonXsuch thatT⊆Y. We have the following definitions.

Definition_3.2. _{For each vertex}v_ofX, definev∗_{to be the unique vertex}

ofTsuch thatG(v,v∗₎_≠_∅_{. That is,}_v_and_v∗_{are in the same}_G_{vertex orbit.} It is clear that ifv is a vertex ofT, thenv∗₌_v_{and, in general, for any two} verticesuandv ofXsuch thatG(u,v)≠∅, we haveu∗₌_v∗_{, and}_G_u _and Gv are conjugate by an element ofG. That is, for every elementbofGu, there

existgofGandaofGvsuch thatb=gag−1.

Definition3.3. For each edgeyofY₀∪Y₁∪Y₂, define[y]to be an element ofG(t(y),(t(y))∗_{). That is,}_[y]_{satisfies the condition that}_[y]((t(y))∗₎₌_t(y), and to be chosen as follows:

[y]=1 ify∈Y0, [y](y)=y¯ ify∈Y2. (3.1)

Define[y]¯ to be the element

[y]¯ =   

[y] ify∈Y0∪Y2,

[y]−1 _if_y_∈_Y 1.

(3.2)

From above we see that[y](y)≠y¯ify∈Y0∪Y1and[y](y)=y¯ify∈Y2.

A group is termed aquasifreegroup if it is a free product of copies ofC_∞ andC2, whereC∞denotes infinite cyclic group andC2a 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 ofC∞and a zero number of copies ofC2;

(2) the group of the presentation x,y,z|z2 ₌₁ _C

∞∗C∞∗C2 is a

quasifree group;

(3) the infinite dihedral groupx,y|x2₌_{1, y}2₌₁ _C

2∗C2is a quasifree

group.

Lemma_3.4. _LetG_,X_,Y_{, and}T _{be as above such that the stabilizer of each}

vertex ofXis trivial. Then,Gis a quasifree group.

Proof. _{By [4, Theorem 3.6],}G_{has the presentation}

Gv,y,x|relGv, Gm=Gm¯, y·[y]−1Gy[y]·y−1=Gy,

x·Gx·x−1=Gx, x2=[x]2

(3.3)

via the mapGv→Gv,y→[y], andx→[x]wherev∈V (T ),m∈Y0,y∈Y1,

andx∈Y2.

Since the stabilizer of each vertex ofX is trivial,Gv,Gm, Gy, andGxare

trivial for allv∈V (T ),m∈Y0,y∈Y1, andx∈Y2. This implies that[x]2=1

for allx∈Y2. Then,Ghas the presentationy,x|x2=1, wherey∈Y1and

(4)

This implies thatGis a quasifree group. This completes the proof.

Corollary3.5. LetG, X, Y, and T be as above, andH a subgroup ofG

such thatH∩Gv is trivial for allv∈V (X). ThenHis a quasifree group.

Definition_3.6. _{For each edge}y_ofY_{, define the following.}

(1) Define −y to be the edge −y =[y]−1_(y)_if _o(y)_∈_{V (T ), otherwise} −y=y.

(2) Define+yto be the edge+y=[y](−y). It is clear thatt(−y)=(t(y))∗_,_o(₊_y)₌_(o(y))∗_,_G

−y≤G(t(y))∗, andG₊y≤

G(o(y))∗. Moreover, ify∈Y0∪Y2, thenG−y=G+y=Gy.

(3) Defineφy to be the mapφy:G−y→G+ygiven byφy(g)=[y]g[y]−1.

It is clear thatφy is an isomorphism.

(4) Defineδyto be the elementδy=[y][y].¯

It is clear thatδy=1 ify∈Y0∪Y1, andδy=[y]2ify∈Y2. Consequently,

δy∈Gy,δy¯=δy, andφy(δy)=δy.

Definition_3.7. _{By a word}w _ofG, we mean an expression of the form

w=g0·y1·g1·y2·g2···yn·gn,n≥0,yi∈E(Y ), fori=1,2,...,nsuch that

(1) g0∈G(o(y1))∗,

(2) gi∈G(t(yi))∗fori=1,2,...,n,

(3) (t(yi))∗=(o(yi+1))∗fori=1,2,...,n−1.

Ifw=1, thenwis called a trivial word ofG.

Letwbe the word defined above. We have the following concepts:

(a) wis called reduced ifwcontains no expression of the formyi·giyi−1

ifgi∈G−yi, oryi·gi·yiifgi∈Gyi andG(yi,y¯i)≠∅;

(b) we defineo(w)=(o(y1))∗ andt(w)=(t(yn))∗. Ifo(w)=t(w)=v,

thenwis called a closed word ofGof typev;

(c) the value of w denoted by [w] is defined as the element [w] = g0[y1]g1[y2]g2···[yn]gnofG;

(d) nis called the length ofwand is denoted by|w| =n;

(e) the inverse ofwdenoted byw−1_{is defined as the word}_w−1₌_g−1

n ·y¯n·

δ−1

yng−

1

n−1···g−21·y¯2·δ−y12g −1

1 ·y¯1·δ−y11g −1 0 ofG.

It is clear that[w−1_]₌_[w]−1_but_(w−1₎−1_≠_w_if_w_{contains an edge}_y_(say)

such thatG(y,y)¯ ≠ϕ. Otherwise,(w−1₎−1₌_w.

Proposition3.8. Every element ofGis the value of a closed and reduced

word ofG. Moreover, ifwis a nontrivial closed and reduced word ofG, then

[w]is not the identity element ofG. Moreover, ifw1andw2are two closed and reduced words ofGsuch thatw1andw2are of the same type and of the same value, then|w1| = |w2|.

Proof. _{See [5, Corollary 3.6].}

(5)

Theorem_4.1. _LetG_{be a group acting on a tree}X_{with inversions such that}

each edge stabilizer is malnormal inG. Letgbe a nontrivial element ofGand

C(g)the centralizer ofginG. Then,

(i) ifgis in a vertex stabilizer, thenC(g)is in that vertex stabilizer;

(ii) ifgis not in any vertex stabilizer, thenC(g)is an infinite cyclic subgroup ofGifgdoes not transfer an edge ofXto its inverse. Otherwise,C(g)

is a finite cyclic subgroup ofGof order2.

Proof. (i) Letvbe a vertex ofXsuch thatg∈G_v. We need to show that

C(g)is contained inGv. Letfbe an element ofGsuch thatf g=gf. We need

to show thatfis inGv. We consider two cases.

Case1(gis inG_x, wherexis an edge ofXsuch thatvis an end ofx). Since the edge stabilizer for each edge ofXis malnormal inG, thenGxis malnormal

inGv. This implies thatf is inGx. Consequently,f is inGv or, equivalently,

C(g)is contained inGv.

Case₂(g_{is not in any edge stabilizer of}G). _{Then, there exists a unique}

vertexv∗ ofT such thatG(v,v∗)≠∅, that is,v andv∗ are in the sameG vertex orbit. Then, there exista∈Gandb∈Gv∗ such thatg=aba−1_{. This}

implies thatv=a(v∗₎_and_a−1_{f ab}₌_ba−1_{f a. Let}_h₌_a−1_{f a. Since}_g_∈_G

xfor

allx∈E(X),t(x)=v, thereforeh∈G−y, for ally∈E(Y )such that(t(y))∗=

v. ByProposition 3.8, there exists a reduced wordw=g0·y1·g1···yn·gn

ofGsuch thatwis of typev∗ _{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 ofGof value 1, the identity element ofG. Therefore, byProposition 3.8, the wordw·h·w−1_·_h−1₌_g

0·y1·g1···yn·gnhgn−1·y¯n·δ−y1ng−

1

n−1···g−21·y¯2·

δ−1

y2g −1

1 ·y¯1·δ−y11g −1

0 h−1is not reduced. The only way that the indicated word

can fail to be reduced is that gnhg−n1 ∈G−yn. Replacing a subword of the form yi·gi·y¯i if gi∈G−yi byφyi(giδyi), or replacing a subword of the formyi·gi·yiifgi∈Gyi andG(yi,y¯i)≠∅, byφyi(giδyi), we see that each Li=gn−iφyn−i+1(Li−1)g−

1

n−iis inG−yn−ifori=1,...,nwith the convention that L0=gnhgn−1andLn=g0φy1(Ln−1)g−

1

0 h−1=1. Thenh=g0φy1(Ln−1)g−

1 0 .

Since(o(y1))∗=v∗,φy1(Ln−1)∈G+y1≤Gv∗, andg0∈Gv∗, thenh∈Gv∗. This implies thata−1_{f a}_∈_G

v∗. Therefore,f∈aGv∗a−1=Gv. Consequently

C(g)is contained inGv.

(ii) Now, suppose thatgis not in any vertex stabilizer ofG. Then,C(g)has trivial intersection with each vertex stabilizer ofG. Ifa≠1 is inC(g), andais in a vertex stabilizerGv ofG, for the vertexvofX, thengis inC(a)and, by

above,gis inGv. This contradicts the assumption thatgis not in any vertex

(6)

group of order 2. Otherwise,C(g)is an infinite cyclic group. This completes the proof.

Definition_4.2. _Letn_{be a positive integer and}g_{a nontrivial element of}

the groupH. We say thatghas at mostnth root if wheneverg=an₌_bn_{, for}

a,binH, thena=b.

In the next corollaries, the groupGsatisfies the hypothesis ofTheorem 4.1.

Corollary4.3. Any element ofGthat is not in any vertex stabilizer ofG

has at mostnth root.

Proof. _Letg,a, andb _{be elements of}G_{such that} g_{is not in any vertex}

stabilizer ofG, andg=an₌_bn_{. 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.

Sincega=agandgb=bg, thenaandbare inC(g). Then, it is clear that g=an₌_bn_{implies that}_a₌_{b. This completes the proof.}

Corollary _4.4. _Letg _{be an element of} G_{. Then,} g _{is not in any vertex}

stabilizer ofGif and only ifgn_{is not in any vertex stabilizer of}_G_{, where}_n_{is a}

positive integer.

Proof. _Sincegn_{commutes with}g, thengn_{is in}C(g)_{which, by}_Theorem

4.1, is not in any vertex stabilizer ofGand the result follows. This completes the proof.

Corollary4.5. Letfandgbe two elements ofG, andmandntwo positive

integers such thatf andgare not in any vertex stabilizer of Gandfm_gn₌

gn_fm_{. Then}_{f g}₌_gf_.

Proof. _From_{Corollary 4.4, we get}

fm_gn₌_gn_fm _⇒_fm_gn_f−m₌_gn

⇒fm_gf−mn

=gn

⇒fm_gf−m₌_g

⇒fm₌_gfm_g−1

⇒fm₌_{gf g}−1m

⇒f=gf g−1 ⇒f g=gf .

(4.1)

This completes the proof.

Corollary_4.6. _Letf _andg_{be two elements of}G_{such that}f _andg_are

(7)

Proof. _By_{Theorem 4.1,}C(f )_and C(g)_{are cyclic subgroups of}G. Then,

there exist two elementsaandbofGsuch thataandbare not in any vertex stabilizer ofG, C(f )= a, andC(g)= b. It is clear that ifa∈C(g), then C(f )=C(g). There exist two positive integersm and nsuch that f =am

and g=bn_{. Since}_{f g}₌_gf_,_am_bn₌_bn_am_{. Then,}_{Corollary 4.5}_{implies that}

ab=ba. This implies thatabn₌_bn_{a. Then}_a_∈_C(bn₎₌_{C(g). This completes}

the proof.

5. Applications. This section is an application ofTheorem 4.1and its corol-laries. Free groups, free product of groups, free product of groups with amal-gamation 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 amalga-mation subgroup are special cases of tree product of groups and free groups and HNN groups are special cases of quasi-HNN groups.

Proposition5.1. LetG=∗_i_∈_I(A_i,U_jk=U_kj)be a nontrivial tree product

of the groupsAi,i∈I, such thatUijare malnormal subgroups ofG. Letgbe a

nontrivial element ofGandC(g)be the centralizer ofginG. Then,

(i) C(g)is in a conjugate ofAifor somei,i∈I, ifgis in a conjugate ofAi;

(ii) ifgis not in a conjugate ofAi, for alli∈I, thenC(g)is an infinite cyclic

group andghas at mostnth root.

Proof. _{By [6], there exists a tree}X_{on which}G_{acts without inversions such}

that any tree of representatives for the action ofGonXequals the correspond-ing fundamental domain for the action ofGonX, and for every vertexuofX and every edgexofX,Guis isomorphic toAi,i∈I, andGxis isomorphic to

Uikfor somei,kinI. Moreover,Gcontains no invertor elements. Therefore,

byTheorem 4.1andCorollary 4.3, the proof ofProposition 5.1 follows. This completes the proof.

Corollary_5.2. _LetG=A∗

CBbe a free product of the groupsAandBwith

amalgamation subgroupCsuch thatCis a malnormal subgroup ofG. Letgbe a nontrivial element ofGandC(g)the centralizer ofginG. Then,

(i) C(g)is in a conjugate ofAorBifgis in a conjugate ofAorB;

(ii) ifgis not in a conjugate ofAorB, thenC(g)is an infinite cyclic group andghas at mostnth root.

Corollary_5.3. _LetG=A∗B_{be a free product of the groups}A_andB_{. Let}

gbe a nontrivial element ofGandC(g)the centralizer ofginG. Then,

(i) C(g)is in a conjugate ofA orB ifgis in a conjugate ofAor is in a conjugate ofB;

(8)

Proposition_5.4. _LetG∗_{be the quasi-HNN group}

G∗₌ _G,t_i_,t_j_|_{relG, t}_i_A_i_t−1

i =Bi, tjCjt−j1=Cj, tj2=cj, i∈I, j∈J (5.1)

such thatAi,Bi, andCj,i∈Iandj∈J, are malnormal subgroups ofG∗. Letg

a nontrivial element ofG∗_and_C(g)_{the centralizer of}_g_in_G∗_{. Then,} (i) C(g)is in a conjugate ofGifgis in a conjugate ofG;

(ii) ifgis not in a conjugate ofG, thenC(g)is an infinite cyclic group or a finite cyclic group of order2andghas at mostnth root.

Proof. _{By [8, Lemma 5.1], there exists a tree}X _{on which} G∗ _{acts with}

inversions such thatG∗ is transitive on V (X), and for every vertexv ofX and every edgex of X, G∗

v is isomorphic toG and G∗x is isomorphic toAi,

i∈I, or isomorphic toCj,j∈J. Moreover,G∗contains the invertor elements

conjugate to an elementtj, j ∈J. Therefore, byTheorem 4.1, the proof of

Proposition 5.4follows. This completes the proof.

Corollary5.5. LetG∗be the HNN group

G∗₌ _G,t_i_|_rel_{G, t}_i_A_i_t−1

i =Bi, i∈I (5.2)

such thatAiandBi,i∈I, are malnormal subgroups ofG∗. Letga nontrivial

element ofG∗_and_C(g)_{the centralizer of}_g_in_G∗_{. Then,} (i) C(g)is in a conjugate ofGifgis in a conjugate ofG;

(ii) ifgis not in a conjugate ofG, thenC(g)is an infinite cyclic group and

ghas at mostnth root.

Corollary 5.6. If g is a nontrivial element of a free group F, then the

centralizerC(g)ofginF is an infinite cyclic group andghas at mostnth root.

Acknowledgment. _{The author would like to thank the referee for his}

sin-cere evaluation and constructive comments which improved the paper consid-erably.

References

[1] A. Karrass and D. Solitar,The free product of two groups with a malnormal amal-gamated subgroup, Canad. J. Math.23(1971), 933–959.

[2] M. I. Khanfar and R. M. S. Mahmood,On quasi HNN groups, J. Univ. Kuwait Sci.29 (2002), no. 2, 13–24.

[3] W. Magnus, A. Karrass, and D. Solitar,Combinatorial Group Theory, Presentations of Groups in Terms of Generators and Relations, Dover Publications, New York, 1976.

[4] R. M. S. Mahmood,Presentation of groups acting on trees with inversions, Proc. Roy. Soc. Edinburgh Sect. A113(1989), no. 3-4, 235–241.

[5] ,The normal form theorem of groups acting on trees with inversions, J. Univ. Kuwait Sci.18(1991), 7–16.

(9)

[7] R. M. S. Mahmood and M. I. Khanfar,On invertor elements and finitely generated subgroups of groups acting on trees with inversions, Int. J. Math. Math. Sci. 23(2000), no. 9, 585–595.

[8] ,Subgroups of quasi-HNN groups, Int. J. Math. Math. Sci.31(2002), no. 12, 731–743.

R. M. S. Mahmood: College of Education and Basic Science, Ajman University of Sci-ence and Technology, Abu Dhabi, United Arab Emirates