OF GROUPS
P.C.WONG, C.K.TANG, AND H.W.GAN
Abstract. In this note we shall prove a characterization for the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups with infinite cyclic edge subgroups to be weakly potent. We shall show also that all one-relator groups with non-trivial center are weakly potent.
1. Introduction.
In this note, we shall prove a characterization for the fundamental groups of graphs of polycyclic-by-finite groups and free-by-finite groups with infinite cyclic edge subgroups to be again weakly potent. Let F be a graph and T be a maxi- mal subtree of F. The fundamental group G of the graph F of groups with in- finite cyclic edge subgroups can be considered as an HNN extension of the form G=hA, t1, . . . , tn;t−1i hiti =ki, i= 1, . . . , niwhereA is a tree product of the ver- tex groups according to the maximal subtree T and the hi, ki are in the vertex groups. Since one of the simplest type of HNN extensions, the Baumslag-Solitar group,hh, t;t−1h2t=h3iis not even residually finite (See G. Baumslag and Solitar [3]), the residual properties of fundamental groups of graphs are difficult to deter- mine. Shirvani in [13] first gave conditions for the residual finiteness of fundamental groups of graphs. Recently, Raptis and Varsos [12], Varsos [16] proved the residual nilpotence and subgroup separability of fundamental groups of graphs where the edge group have finite index in the containing vertex groups. More recenty, Kim [8]
obtained characterizations for the fundamental groups of graphs of certain cyclic subgroup separable groups with infinite cyclic edge subgroups to be again cyclic subgroup separable.
2000Mathematics Subject Classification. Primary 20E06, 20E26 20E08; Secondary 20F05.
Key words and phrases. fundamental groups of graphs of groups, weakly potent, cyclic sub- group separable, HNN extensions, tree products, polycyclic-by-finite groups, free-by-finite groups.
1
Weak potency is a strong form of residual finiteness and was first introduced by Evans [4] with the name regular quotients and he showed that free groups and finitely generated torsion-free nilpotent groups are weakly potent. Later, Tang [15] independently defined weak potency and he proved that finite extensions of free groups and finitely generated torsion-free nilpotent groups are weakly potent.
Evans [4] used weak potency to show the cyclic subgroup separability of certain generalised free products while more recently Kim and Tang [9] and Tang [15] used it to determine the conjugacy separability of certain generalised free products of conjugacy separable groups. It is known that polycyclic-by-finite groups and free- by-finite groups are weakly potent (Evans [4], Tang [15]).
A fundamental group of a graphF of groups can be described as follows: (see Kim [8]) Let F = (V, E) be a graph whereV is a set of vertices and E is a set of edges. To each vertexvinV, we assign a groupGv. To each edgeeinE, we assign a group Ge together with monomorphismsαe andβe embeddingGe into the two vertex groups at the end ofe. Then for a maximal subtreeT ofF, the fundamental group of the graphF of groupsGvamalgamating the edge subgroupsGeis defined to be the group generated by all the generators of the vertex groups and additional generatorstefor each edge e∈E. The defining relations are given by the defining relations of all the vertex groups together with the relations te−1(geαe)te =geβe
for eachgeofGewherete= 1 ifeis an edge ofT. Each of the subgroupsGeαeand Geβe is called an edge subgroup in its containing vertex group. It is well known that the fundamental group of a graph of groups is independent from the choice of the maximal subtree (Serre [11]). In particular if the graphF is a tree, then the fundamental group of graph of groups is called a tree product.
Our main result is a characterization for the fundamental group of a graph of weakly potent groups with infinite cyclic edge subgroups to be weakly potent (Theorem 5.1). We then apply this result to graphs of polycyclic-by-finite groups and free-by-finite groups (Theorem 5.2). We prove Theorem 5.1 in two parts. First we prove a characterization for certain HNN extensions of weakly potent groups
with infinite cyclic associated subgroups to be weakly potent (Theorem 3.5). Then we show that the tree products of weakly potent groups with infinite cyclic edge subgroups are again weakly potent (Theorem 4.6). From these results we can also show that all one-relator groups with non-trivial center are weakly potent (Theorem 6.1).
The notations used here are standard. In addition, the following will be used.
LetGbe a group.
N /fGmeansN is a normal subgroup of finite index inG.
G=ht, K;t−1At=B, ϕidenotes the HNN extension where K is the base group, A, B are the associated subgroups andϕis the associated isomorphism
ϕ:A−→B.
Ifx∈G=ht, K;t−1At=B, ϕiis reduced, we shall express xin the form x=x0te1x1· · ·xn−1tenxn
wherex0, xi∈K andei=±1,1≤i≤n.
G=AH∗B denotes the generalised product of the groupsA andB amalgamating the subgroup H.
Ifx∈AH∗B is reduced, we shall express xin the form x=a1b1. . . anbn
whereai∈A\H andbi∈B\H for 1≤i≤n.
kxkwill denote the usual reduced length ofxin the HNN extension or generalised free product G.
2. Preliminaries.
We begin with two definitions.
Definition 2.1. (Tang [15]) A groupGis called weakly potent if for any element xof infinite order inG, we can find a positive integer rwith the property that for every positive integer n, there exists a normal subgroup Mn of finite index in G such thatxMnhas order exactlyrn.
Similarly a group G is called potent if for any elementxof infinite order in G and every positive integern, there exists a normal subgroupMnof finite index in Gsuch thatxMnhas order exactlyn.
From the definition above, potency is a much stronger property than weak po- tency. Stebe [14] first proved that free groups are potent (before Evan introduced weak potency) but he did not give it a name. Indeed Stebe proved that ifn is a power of a prime q then the factor group G/Nn can be chosen to be a q-group.
(Stebe acknowledged that the proof of this result was suggested by D.S.Passman.) Later Allenby and Tang independently introduced potency in [2] and they used it to prove proved finite extensions of certain generalised free products. In [1] Allenby showed that cyclically pinched one-relator groups are potent.
Definition 2.2. A groupG is called H-separable for the subgroup H if for each x∈G\H, there existsN /fGsuch thatx /∈HN.
Gis termed πc ifGishhi-separable for every cyclic subgroup hhi.
Free-by-finite groups and polycyclic-by-finite groups are weakly potent andπc. (see Evans [4], Tang [15]) Indeed certain HNN extensions of polycyclic-by-finite groups with central associated subgroups areπcand even subgroup separable. (see [17], [18])
3. HNN extensions of weakly potent groups
In this section we give conditions for an HNN extension of weakly potent group with cyclic associated subgroups to be weakly potent.
Lemma 3.1. LetG=ht, K;t−1At=B, ϕi whereK is finite. ThenGis free-by- finite (see [5]) and hence weakly potent and πc (see [4], [15]).
Theorem 3.2. Let G =ht, A;t−1ht = ki where A is a weakly potent group and hhi,hkiare infinite cyclic subgroups ofAsuch thathhi ∩ hki 6= 1. Suppose thatAis hhi-separable and hki-separable. ThenG is weakly potent if and only ifhm =k±m for somem >0.
Proof. First we note that since A is weakly potent, we can find positive integers r1, r2 such that for each positive integer n, there exist P /f A, Q /f A such that P∩ hhi=hhr1ni, Q∩ hki=hhr2ni.
Suppose that hm =k±m for some m >0. Let xbe an element of infinite order inG.
Case 1. kxk= 0, that is,x∈A. From above, let N1/fA, N2/fAbe such that N1∩ hhi=hhr1r2mi, N2∩ hki=hkr1r2mi. Let Q=N1∩N2. ThenQ /fA and Q∩ hhi=hhr1r2mi=hkr1r2mi=Q∩ hki.
SupposeQ∩ hxi=hxαifor some integerα. SinceAis weakly potent, we can find a positive integerr such that for each positive integern, there exists P /f Asuch thatP∩ hxi=hxrαni. LetN =P∩Q. ThenN /fAis such thatN∩ hxi=hxrαni andN∩ hhi=N∩ hki.
Now we form ¯G=ht,A;¯ t−1¯ht= ¯kiwhere ¯A=A/N,¯h=hN,k¯=kN. Clearly G¯ is a homomorphic image ofG. Since ¯Ais finite, ¯Gis residually finite by Lemma 3.1 and hence there exists ¯M /fG¯such that ¯x,x¯2, . . . ,x¯rαn−1∈/M¯. Thus ¯xM¯ has order exactly rαnin the finite group ¯G/M¯. Let M be the preimage of ¯M inG.
ThenxM has order exactlyrαninG/M and we are done.
Case 2. kxk ≥1. Letx=x0te1x1te2. . . tenxnwherexi∈Aandn≥1. SinceAis hhi-separable andhki-separable, there existsM /fA such thatxi∈ hhiM/ if xi∈ hhi/ andxi∈ hkiM/ ifxi∈ hki. Suppose/ M∩ hhi=hhα1iandM∩ hki=hkα2i for some integers α1, α2. Letα=α1α2. From above letN1/fA, N2/fA be such thatN1∩ hhi=hhαr1r2miandN2∩ hki=hkαr1r2mi. LetN =M∩N1∩N2. Then N /fAandN∩ hhi=hhαr1r2mi=hkαr1r2mi=N∩ hki. As in Case 1, we form ¯G.
Then ¯xis reduced in ¯Gand k¯xk=kxk. It follows that ¯xhas infinite order in ¯G.
Since ¯Ais finite, ¯Gis weakly potent by Lemma 3.1 and our result follows.
Conversely, suppose thatGis weakly potent. Sincehhi ∩ hki 6= 1, it follows that hm=kp wherep >0. SinceGis weakly potent, the subgroupG1=ht, h;t−1hpt= hmiis weakly potent. Then there exists a positive integerrwith the property that
for every positive integer n, we can find N /fG1 such that hN has order exactly rn. We choose n=|p||m|. Then |hm |=r|p| and |hp |=r |m|. Since ¯hp is conjugate to ¯hm in ¯G1=G1/N,r|m|=r|p| which implies that|m|=|p|.
By using Theorem 3.2, we can easily obtain a characterization for the Baumslag- Solitar groups to be weakly potent.
Theorem 3.3. LetGk,l =ht, a;t−1akt =ali. Then Gk,l is weakly potent if and only if|k|=|l|.
Next we extend Theorem 3.2 to HNN extensions of the formG=hA, t1, . . . , tn; t−1i hiti=ki, i= 1, . . . , ni.
Lemma 3.4. Let G = ht, A;t−1ht = ki where A is a weakly potent group and hhi,hki are infinite cyclic subgroups of A such that hm = k±m for some m > 0.
Suppose that A is hhi-separable and hki-separable. Let a∈ A such that A is hai- separable. Then G=ht, A;t−1ht=kiishai-separable.
Proof. Letx∈G\haibe a reduced element inG.
Case 1. |x|= 0, that is,x∈A. SinceAishai-separable, there existsM /fAsuch thatx /∈ haiM. As in the Case 2 of Theorem 3.2, there existsN /f Asuch that N ⊆M andN∩ hhi=N∩ hki. We form ¯G=ht,A;¯ t−1¯ht= ¯kiwhere
A¯=A/N,¯h=hN,¯k=kN. Clearly ¯Gis a homomorphic image ofG. Since
¯
x /∈ h¯aiand ¯Gisπc by Lemma 3.1, the result now follows.
Case 2. |x|≥1. Letx=x0te11x1te12. . . te1nxnwhere xi∈A andn≥1. Since Ais hhi-separable andhki-separable, there existsM /fA such thatxi∈ hhiM/ if xi∈ hhi/ andxi∈ hkiM/ ifxi∈ hki. Again as in Case 2 of Theorem 3.2, there/ existsN /fAsuch thatN ⊆M andN∩ hhi=N∩ hki. Again we form ¯G. Then
¯
xis reduced in ¯Gandk¯xk=kxk. Since ¯x /∈ h¯aiand ¯Gisπc by Lemma 3.1, again we are done.
Theorem 3.5. LetAbe weakly potent and hi, ki∈Abe elements of infinite order such that hhii ∩ hkii 6= 1 for each i = 1, . . . , n. Suppose A is hhii-separable and hkii-separable. Then G=hA, t1, . . . , tn;t−1i hiti=ki, i= 1, . . . , ni is weakly potent if and only if for eachi= 1, . . . , n,hmi i=ki±mi for somemi >0.
Proof. Suppose that Gis weakly potent. Since hA, ti;t−1i hiti =kiiis a subgroup ofG, then it must be weakly potent. Hence by Theorem 3.2,hmi i =k±mi i for some mi>0.
Conversely, suppose that for each i = 1, . . . , n, hmi i =k±mi i for some mi >0.
Let G1 = hA, t1;t−11 h1t1 =k1i. Then G1 is weakly potent by Theorem 3.2 and G1 is hhii-separable and hkii-separable for each i= 2, . . . , n by Lemma 3.4. Let Gj =hA, t1, . . . , tj;t−1i hiti=ki, i= 1, . . . , ji. Then Gj =hGj−1, tj;t−1j hjtj =kji.
Inductively, we assume that Gn−1 = hA, t1, . . . , tn−1;t−1i hiti = ki, i = 1, . . . , n− 1i is weakly potent and hhni-separable and hkni-separable. Then G = Gn = hGn−1, tn;t−1n hntn=kniis weakly potent by Theorem 3.2.
Since polycyclic-by-finite groups and free-by-finite groups are weakly potent and πc (Evans [4], Tang [15]), we have immediately the following:
Theorem 3.6. LetAbe a free-by-finite or polycyclic-by-finite group andhi, ki∈A be elements of infinite order such that hhii ∩ hkii 6= 1 for each i= 1, . . . , n. Then G=hA, t1, . . . , tn;t−1i hiti=ki, i= 1, . . . , niis weakly potent if and only if for each i= 1, . . . , n,hmi i=ki±mi for somemi>0.
Note that in Theorem 3.5, ifAis a tree product wherehi andkimay not be in the same vertex group, thenGcan be considered as a fundamental group of a graph of groups amalgamating cyclic edge subgroups. In order to do this, we next prove the weak potency of tree products of weakly potent groups with infinite cyclic edge subgroups in the next section.
4. Tree products of weakly potent groups
In this section we will show that tree products of finitely many weakly potent groups amalgamating infinite cyclic subgroups are weakly potent.
Lemma 4.1. Let G=G1
∗
HG2 where G1, G2 are finite. Then G is free-by-finite (see [6]) and hence weakly potent and πc (see [4], [15]).
Theorem 4.2. LetG=G1
∗
HG2 where G1, G2 are weakly potent and H =hhiis infinite cyclic. Suppose G1, G2 arehhi-separable. ThenGis weakly potent.
Proof. First we note that since G1, G2 are weakly potent, we can find positive integers r1, r2 such that for each positive integern, there exist P /fG1, Q /f G2
such thatP∩ hhi=hhr1ni, Q∩ hhi=hhr2ni. Letg be an element of infinite order inG.
Case 1. kgk ≤1, that is,g∈G1∪G2. WLOG, assume g∈G1. From above let N1/fG1be such that N1∩ hhi=hhr1r2i. SupposeN1∩ hgi=hgsifor some positive integer s. By the weak potency ofG1, we can find a positive integerr such that for each positive integern, there existsN2/fG1 such that
N2∩ hgi=hgrsni. Let N =N1∩N2. ThenN /fG1,N∩ hgi=hgrsniand N∩ hhi=hhr1r2tifor some positive integert. Let M /fG2be such that M ∩ hhi=hhr1r2ti. Now we form ¯G= ¯G1
∗
H¯G¯2where ¯G1=G1/N, ¯G2=G2/M and ¯H=hhiN/N =hhiM/M. Clearly ¯Gis a homomorphic image ofG. Let ¯g denote the image ofg in ¯G. Then ¯ghas order exactly rsnin ¯G. Since ¯Gis residually finite by Lemma 4.1, there exists ¯P /fG¯such that ¯g, . . . ,¯grsn−1∈/P.¯ LetP be the preimage of ¯P inG. ThenP /fGandgP has order exactlyrsnin G/P and we are done.
Case 2. kgk>1, that is,g /∈G1∪G2. WLOG, assume g=a1b1. . . anbnwhere ai∈G1\hhiandbi∈G2\hhifor alli. SinceG1,G2 arehhi-separable, there exist N1/fG1,M1/fG2 such thatai∈ hhiN/ 1 andbi∈ hhiM/ 1 for alli. Suppose N1∩ hhi=hhs1iandM1∩ hhi=hhs2ifor some positive integerss1 ands2. Let
N2/fG1,M2/fG2 be such thatN2∩ hhi=hhr1r2s1s2i=M2∩ hhi. Let
N =N1∩N2 andM =M1∩M2. ThenN /fG1, M /fG2andN∩ hhi=M∩ hhi.
As in Case 1, we form ¯G. Thenk¯gk=kgkand hence ¯ghas infinite order in ¯G. By Lemma 4.1, ¯Gis weakly potent and the result follows.
To extend Theorem 4.2 to a tree product, we need the next few lemmas.
Lemma 4.3. (Kim [7]) LetG=G1∗
HG2. Suppose that (a) G1, G2 areH-separable;
(b) for eachR/fH, there existN /fG1, M /fG2such thatN∩H=M∩H⊆R.
LetK be a subgroup ofG1 and G1 isK-separable. Then Gis K-separable.
The next lemma can easily be derived from Lemma 4.3.
Lemma 4.4. Let G = G1
∗
HG2 where H = hhi is infinite cyclic. Suppose that G1, G2 are weakly potent and hhi-separable. LetK be a subgroup of G1 and G1 is K-separable. Then GisK-separable.
Lemma 4.4 can be extended to a tree product with the additional condition that the groupGis weakly potent.
Lemma 4.5. Let G = hG1, G2, . . . , Gn;aij = ajii be a tree product of G1, G2, . . . , Gn, amalgamating the infinite cyclic subgroups haiji of Gi and hajii of Gj. Suppose Gis weakly potent and eachGi ishaiji-separable. LetK be a subgroup of Gr and Gr is K-separable. Then Gis K-separable.
Proof. We use induction onn. The casen= 2 follows from Lemma 4.4. Now, let n > 2. The tree product G has an extremal vertex, say Gn, which is joined to a unique vertex, sayGn−1. The subgroup ofG generated byG1, G2, . . . , Gn−1 is just their tree product. LetG0denote this subgroup. ThenG=hG0, Gn;a(n−1)n= an(n−1)i. By the inductive hypothesis,G0isha(n−1)ni-separable and by assumption,
Gnishan(n−1)i-separable. Furthermore, by assumptionGis weakly potent. Hence G0is weakly potent andGnis weakly potent.
Case 1. K⊆G0. By inductive hypothesis,G0is K-separable and we are done by Lemma 4.4.
Case 2. K⊆Gn. By assumption,GnisK-separable and we are done by Lemma 4.4.
Now, Theorem 4.2 can be extended to a tree product as follows:
Theorem 4.6. Let G = hG1, G2, . . . , Gn;aij = ajii be a tree product of G1, G2, . . . , Gn, amalgamating the infinite cyclic subgroups haiji of Gi and hajii of Gj. Suppose eachGi is weakly potent and haiji-separable. Then Gis weakly potent.
Proof. We use induction on n. The case n = 2 follows from Theorem 4.2. Now, letn >2. As in Lemma 4.5, we write G =hG0, Gn;a(n−1)n=an(n−1)iwhere G0 is the tree product generated byG1, G2, . . . , Gn−1. By inductive hypothesis,G0is weakly potent. Hence G0 is ha(n−1)ni-separable by Lemma 4.5. Furthermore by assumption, Gn is weakly potent and Gn is han(n−1)i-separable. Therefore G is
weakly potent by Theorem 4.2.
Corollary 4.7. Let G1, G2, . . . , Gn be free-by-finite groups or polycyclic-by-finite groups. Let G = hG1, G2, . . . , Gn;aij = ajii be a tree product of G1, G2, . . . , Gn, amalgamating the infinite cyclic subgroups haiji of Gi and hajii Gj. Then G is weakly potent.
5. Fundamental groups of graphs of weakly potent groups From Theorem 3.5 and Theorem 4.6, we obtain our main results.
Theorem 5.1. LetAv be weakly potent groups. LetG be a fundamental group of a graph of the groups Av amalgamating cyclic edge subgroups, presented by G =
hA, t1, . . . , tn;t−1i hiti = ki, i = 1, . . . , ni where A is a tree product of the groups Av according to a maximal subtree of the graph and where hhii ∩ hkii 6= 1 for each i = 1, . . . , n. Then G is weakly potent if and only if for each i = 1, . . . , n, hmi i =k±mi i for somemi>0.
Theorem 5.2. Let Av be free-by-finite or polycyclic-by-finite groups. Let Gbe a fundamental group of a graph of the groupsAv amalgamating cyclic edge subgroups, presented byG=hA, t1, . . . , tn;t−1i hiti=ki, i= 1, . . . , niwhereA is a tree product of the groupsAvaccording to a maximal subtree of the graph and wherehhii∩hkii 6=
1for eachi= 1, . . . , n. ThenGis weakly potent if and only if for eachi= 1, . . . , n, hmi i =k±mi i for somemi>0.
6. One-relator groups with non-trivial center
Next we apply Theorem 3.2 and Theorem 4.6 to show that all one-relator groups with non-trivial center are weakly potent.
Theorem 6.1. Let G be a one-relator group with non-trivial centre. Then G is weakly potent.
Proof. First suppose that the abelianisation of Gis not free abelian of rank two.
Then by Pietrowski [10, Theorem 1],Ghas a presentation of the form
ha1, a2, . . . , am;ap11=aq21, ap22 =aq32, . . . , apm−1m−1=aqmm−1i wherem, pi, qi≥2 and (pi, qj) = 1 fori > j. ClearlyGis a tree product of infinite cyclic groups and hence Gis weakly potent by Theorem 4.6.
Now suppose that the abelianisation ofGis free abelian of rank two. Again by Pietrowski [10, Theorem 3],Ghas a presentation of the form
ht, a1, a2, . . . , am;t−1a1t=am, ap11 =aq21, ap22 =aq32, . . . , apm−1m−1 =aqmm−1i
wherem, pi, qi≥2 and (pi, qj) = 1 fori > j such thatp1p2. . . pm−1= q1q2. . . qm−1. ThenG=ht, B;t−1a1t=amiis an HNN extension where
B=ha1, a2,· · ·, am;ap11 =aq21, ap22=aq32,· · ·, apm−1m−1 =aqmm−1iand aδ1=aδm where δ=p1p2. . . pm−1=q1q2. . . qm−1. NowB is a tree product of infinite cyclic groups and hence is weakly potent by Theorem 4.6. By Theorem 2.1 of Kim [6], B isπc and henceB is ha1i-separable andhami-separable. ThereforeGis weakly potent by Theorem 3.2.
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Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia
E-mail address:[email protected]
Department of Computer Science and Mathematics, Faculty of Information and Com- munication Technology, Universiti Tunku Abdul Rahman, 46200 Petaling Jaya, Selan- gor, Malaysia
E-mail address:[email protected]
Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia
