The Cyclic Subgroup Separability of Certain HNN Extensions

BULLETINof the Malaysian Mathematical Sciences Society

http://math.usm.my/bulletin

Bull. Malays. Math. Sci. Soc. (2)29(1) (2006), 111–117

The Cyclic Subgroup Separability of Certain HNN Extensions

1P.C. Wong and ²K.B. Wong

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

1[email protected],²[email protected]

Abstract. In this note we give characterisations for certain HNN extensions with central associated subgroups to be cyclic subgroup separable. We then apply our results to HNN extensions of polycyclic-by-finite groups and Fuch- sian groups.

2000 Mathematics Subject Classification: 20E06, 20E26

Key words and phrases: HNN extensions, cyclic subgroup separable, polycyclic- by-finite groups, free-by-finie groups, Fuchsian groups, abelian groups.

1. Introduction

A group G is called cyclic subgroup separable (orπc for short) if for each cyclic subgroupH and x∈G\H, there exists a normal subgroup N of finite index inG such thatx /∈HN. Clearly a cyclic subgroup separable group is residually finite.

The concept of cyclic subgroup separability was introduced by Stebe [14] in 1968 and he used it to prove the residual finiteness of a class of knot groups.

Many classes of groups, including the free groups and the polycyclic-by-finite groups are cyclic subgroup separable ([7], [10]). Also the finite extension of a cyclic subgroup separable group is again cyclic subgroup separable (Stebe [14]).

On the other hand, the cyclic subgroup separability of HNN extensions are not much known. For example, the HNN extensionhh, t;t⁻¹ht=h²iis residually finite but is not cyclic subgroup separable (see [1]) while another HNN extension, the Baumslag-Solitar group,hh, t;t⁻¹h²t=h³iis not even residually finite (see [5]).

Kim [8] and Kim and Tang [9] gave characterisations for HNN extensions of cyclic subgroup separable groups with cyclic associated subgroups to be again cyclic subgroup separable. They then apply their results to give characterisations for the HNN extensions of a finitely generated abelian group with cyclic associated subgroups to be cyclic subgroup separable and show that certain HNN extensions of finitely generated torsion-free nilpotent groups with cyclic associated subgroups to be again cyclic subgroup separable.

Received:November 30, 2004;Revised: October 3, 2005.

Andreadakis, Raptis and Varsos in a series of papers [2], [3], [4] and [11], gave characterisations for HNN extensions of a finitely generated abelian group to be residually finite. Some of these results were extended by Wong and Tang [15] to characterisations for HNN extensions of finitely generated abelian groups to be cyclic subgroup separable.

In this note we investigate the cyclic subgroup separability of HNN extensions with central associated subgroups. More precisely, let G = ht, A;t⁻¹Ht = K, ϕi denote an HNN extension where A is the base group, H, K are the associated subgroups and ϕ:H −→K is the associated isomorphism. We shall show that if G=ht, A;t⁻¹Ht=K, ϕiis an HNN extension whereHandKare subgroups in the center of A, H 6=A6=K and Ais subgroup separable, then Gis cyclic subgroup separable if and only if its subgroup (HNN extension) G1 = ht, HK;t⁻¹Ht = K, ϕiis cyclic subgroup separable. Thus we are able to use the results of [11] and [15] to give characterisations for HNN extensions of polycyclic-by-finite groups and Fuchsian groups with central associated subgroups to be cyclic subgroup separable.

More importantly, our result shows that the study of the cyclic subgroup sep- arability of HNN extensions with central associated subgroups can be reduced to that of the cyclic subgroup separability of HNN extensions of abelian groups. Thus the characterisations given in the papers [11] and [15] can be applied to these HNN extensions.

The notation used here is standard. In addition, the following will be used for any groupG: N /fGmeansN is a normal subgroup of finite index inG.

2. Preliminaries

Definition 2.1. A group G is called H-separable for the subgroup H if for each x∈G\H, there existsN /fGsuch thatx /∈HN. The group Gis termed subgroup separable ifGisH-separable for every finitely generated subgroupH. The groupG is termed cyclic subgroup separable (orπc for short) ifGis H-separable for every cyclic subgroupH.

It is well known that free groups, polycyclic groups and surface groups are sub- group separable (M. Hall [7], Mal’cev [10], Scott [12]). Since a finite extension of a subgroup separable group is again subgroup separable, polycyclic-by-finite groups and Fuchsian groups (finite extension of surface groups) are subgroup separable.

3. The main results

In this section we will prove our main results, i.e., Theorem 3.2 and Theorem 3.3. To simplify our exposition we will use the termπcinstead of cyclic subgroup separable for the rest of the paper.

The following lemma is an application of a result of Blass and Newman in [6].

Lemma 3.1. Let G=ht, A;t⁻¹Ht=K, ϕibe an HNN extension. If K =A and H 6=A, thenGis notH-separable.

Remark 3.1. In this note we consider the HNN extensionG=ht, A;t⁻¹Ht=K, ϕi where H and K are subgroups in the center ofA. Note that ifH =A=K, then Ais abelian. Furthermore A is normal inGandG/A∼=hti. HenceGis polycyclic

and by [10],Gis subgroup separable and henceπ_c. Now by Lemma 3.1, ifK=A, H 6= A and H is cyclic, then G is not π_c. So we shall only consider the HNN extensions whereH 6=A6=K.

Theorem 3.1. Let G = ht, A;t⁻¹Ht = K, ϕi be an HNN extension. Suppose H and K are subgroups in the center of A and H 6= A 6= K. Let ∆ = {N /f

A; ϕ(N∩H) =N∩K}. ThenGisπ_cif and only if T

N∈∆

N H=H, T

N∈∆

N K=K and T

N∈∆

Nhai=hai,∀a∈A.

Proof. SupposeGis πc. ThenGis residually finite. Suppose A=HK. SinceH and K are in the center of A, the subgroupHK satisfies the nontrivial identity W(x, y) =x⁻¹y⁻¹xy. FurthermoreH *K andK *H because H 6=A=HK 6=

K. Therefore by [13, Theorem 3’], T

N∈∆

N H =H and T

N∈∆

N K =K.

SupposeA6=HK. Letc∈A−HK. Then the subgroup generated bycandHK, i.e., hc, HKi, is abelian and satisfies the nontrivial identity W(x, y) =x⁻¹y⁻¹xy.

FurthermoreHK is properly contained inhc, HKi. Therefore by [13, Theorem 3], T

N∈∆

N H=H and T

N∈∆

N K=K.

So in both cases we have T

N∈∆

N H =H and T

N∈∆

N K =K.

Next we show that T

N∈∆

Nhai=hai. Leta∈Aandx /∈ hai. SinceGisπ_c, there exists M /fGsuch that x /∈Mhai. Note that (M ∩A)∈∆ andx /∈(M ∩A)hai.

This implies that T

N∈∆

Nhai=hai.

The converse follows from [8, Theorem 2.2].

By using Theorem 3.1, we shall prove Theorem 3.2 and Theorem 3.3. Before that, we state Lemma 3.2, whose proof is immediate from Theorem 3.1 and the result of [6].

Lemma 3.2. Let G=ht, A;t⁻¹Ht=K, ϕibe an HNN extension where H andK are subgroups in the center of Aand H 6=A6=K. SupposeG isπc. IfH ⊆K or K⊆H thenH =K.

Theorem 3.2. Let G=ht, A;t⁻¹Ht=H, ϕi be an HNN extension whereH is a finitely generated subgroup in the center ofAandH 6=A. IfA isHⁿhai-separable for everya∈Aand every positive integern, thenGisπc.

Proof. LetN ∈∆ where ∆ ={N /fA; ϕ(N∩H) =N∩H}. By Theorem 3.1, it is sufficient to show that T

N∈∆

N H =H and T

N∈∆

Nhai=hai,∀a∈A.

First we show that T

N∈∆

N H=H. Leta∈A−H. SinceAisH-separable, there exists M_a/_f A such that a /∈M_aH. Note that M_aH /_f A and M_aH ∈ ∆. This implies that T

a∈A−H

M_aH =H and hence T

N∈∆

N H =H.

Next we show that T

N∈∆

Nhai=hai,∀a∈A. But first we construct a subgroup Nn ∈ ∆ for each n ≥2. Let h0, h1, . . . , hm be coset representatives of Hⁿ in H whereh0= 1 andn≥2. SinceAisHⁿ-separable, there existsMn/fA such that

h_i ∈/ M_nHⁿ for 1 ≤ i ≤ m. Let N_n = M_nHⁿ. Then N_n/_f A. We claim that N_n∈∆, that is,N_n∩H=Hⁿ. Clearly we only need to show thatN_n∩H ⊆Hⁿ. Supposea∈(N_n∩H)−Hⁿ. Sincea /∈Hⁿ, we havea=h_i¯hwhereh_i 6= 1 is a coset representative ofHⁿinH and ¯h∈Hⁿ. On the other hand, sincea∈N_n =M_nHⁿ, a=m˜hwherem∈Mn and ˜h∈Hⁿ. But thenhi∈MnHⁿ, a contradiction. Hence Nn∩H ⊆Hⁿ. ThereforeNn∈∆ for eachn≥2.

Leta∈Aandb∈A−hai. We claim thatb /∈Hⁿhaifor somen≥1. Ifb /∈Hhai, then we are done. So we may assume thatb∈Hhai. Letb=ha^j for someh∈H and integer j. Clearly h /∈ H ∩ hai. Since H is finitely generated abelian, there exists an integern≥1, such thath /∈Hⁿ(H∩ hai). This implies thath /∈Hⁿhai, and thereforeb /∈Hⁿhai.

SinceAisHⁿhai-separable, there exists M1/fAsuch thatb /∈M1Hⁿhai. Note that for sufficiently largem,H^m⊆M1. As above, we can constructsNm∈∆ such that Nm∩H = H^m. Let M = Nm∩M1. Then M ∈ ∆ since M ∩H = H^m. Furthermore b /∈ M Hⁿhai. This implies that b /∈ Mhaiand so T

N∈∆

Nhai =hai.

The proof is now completed.

Theorem 3.3. Let G=ht, A;t⁻¹Ht=K, ϕi be an HNN extension where H and K are subgroups in the center of AandH 6=A6=K. Suppose H*K, K *H and AisMhai-separable for every subgroupM /_fHK anda∈A. ThenGisπ_c if and only ifG₁=ht, HK;t⁻¹Ht=K, ϕiisπ_c.

Proof. SupposeGisπ_c. SinceG₁ is a subgroup ofG,G₁ isπ_c. SupposeG₁ isπ_c. SinceH 6=HK 6=K, then by Theorem 3.1, T

M∈∆1

M H=H, T

M∈∆₁

M K=Kand T

M∈∆₁

Mhxi=hxi,∀x∈HK, where ∆₁={M /fHK ; ϕ(M∩ H) =M ∩K}.

Let ∆ ={N /fA ; ϕ(N∩H) = N∩K}. By Theorem 3.1, it is sufficient to show that T

N∈∆

N H =H, T

N∈∆

N K =Kand T

N∈∆

Nhai=hai, ∀a∈A.

We begin by constructing a subgroup N_(b,a,M) ∈ ∆ for each M ∈ ∆1, a ∈ A and b ∈ A−Mhai. Let M ∈∆1, a ∈ A and b ∈ A−Mhai. Let h0, h1, . . . , hm

be coset representatives ofM inHK whereh0= 1. SinceAisM-separable, there exists PM /f A such that hi ∈/ PMM for 1 ≤i≤m. Since A isMhai-separable, there exists P_(b,a)/fA such that b /∈P_(b,a)Mhai. Let P_(b,a,M)=P_(b,a)∩PM and N_(b,a,M) = P_(b,a,M)M. Then N_(b,a,M)/f A and b /∈ N_(b,a,M)hai. Next we claim thatN(b,a,M)∩HK=M. Clearly we need only to show thatN(b,a,M)∩HK ⊆M. Suppose y ∈ (N_(b,a,M)∩HK)−M. Since y /∈ M, then y =him1 where hi 6= 1 is a coset representative of M in HK and m1 ∈ M. On the other hand, since y∈N(b,a,M)=P(b,a,M)M, we havey=pm2where p∈P(b,a,M) andm2∈M. But this implies thathi ∈P(b,a,M)M ⊆PMM, a contradiction. ThusN(b,a,M)∩HK= M. This implies that N_(b,a,M)∩H =M ∩H and N_(b,a,M)∩K =M∩K. Hence N_(b,a,M)∈∆.

Next we show that T

N∈∆

N H = H. Let a ∈ A−H. Suppose a ∈ A−HK. Since Ais HK-separable, there exists Ma/fA such thata /∈MaHK. Note that MaHK /fAandMaHK ∈∆. Supposea∈HK. Sincea∈HK−H, there exists

M ∈ ∆₁ such that a /∈ M H. As above, we can construct a subgroup N_(a,1,M) for M ∈ ∆1, 1∈ A and a∈ A−Mh1i. We claim that a /∈N(a,1,M)H. Suppose a ∈ N_(a,1,M)H. Then a = nh for some n ∈ N_(a,1,M) and h ∈ H. This implies that n ∈ N_(a,1,M)∩HK. But N_(a,1,M)∩HK = M by its construction above.

Hencea∈M H, a contradiction. Thereforea /∈N_(a,1,M)H and thus T

N∈∆

N H=H. Similarly T

N∈∆

N K=K.

Finally we show that T

N∈∆

Nhai = hai,∀a ∈ A. Let a ∈ A and b ∈ A− hai.

Supposeb /∈HKhai. SinceA isHKhai-separable, there exists M /_fA such that b /∈ M HKhai. Note that M HK /_f A and M HK ∈ ∆. Suppose b ∈ HKhai.

Then b=xaⁱ for some x∈HK and integer i. Clearlyx /∈HK∩ hai. Therefore there existsM₁∈∆₁ such that x /∈M₁(HK∩ hai). This implies thatb /∈M₁hai.

As above, we can construct a subgroup N_(b,a,M1)) for M₁ ∈ ∆₁, a ∈ A and b ∈ A−M₁hai. From this construction, we have N_(b,a,M) ∈ ∆ and b /∈ N_(b,a,M1)hai.

Hence T

N∈∆

Nhai=hai. The proof is now completed.

4. Applications

In this section will apply the results in section 3 to HNN extensions of polycyclic- by-finite groups and Fuchsian groups. But first we have the following lemma.

Lemma 4.1. Let A be a group and H and K be isomorphic finitely generated subgroups in the center of A such that ϕ : H −→ K is an isomorphism from H ontoK. Let ∆₁={M /_fHK ; ϕ(M ∩H) = M∩K}. Suppose T

M∈∆1

M H =H

and T

M∈∆1

M K=K. Then there exists N ∈∆₁ such that Nⁿ ∈∆₁ for all n≥1.

Proof. Let i_HK(H) = {b ∈ HK;bⁿ ∈ H for some positive integer n}. Then i_HK(H) is a group and H is of finite index in i_HK(H). Similarly let i_HK(K) = {b∈HK;bⁿ∈Kfor some positive integer n}. Theni_HK(K) is a group andKhas finite index in i_HK(K). Since T

M∈∆1

M H=H, T

M∈∆1

M K=K and H andK are of finite index ini_HK(H) andi_HK(K) respectively, there existsN ∈∆₁ such that N∩i_HK(H) =N∩H andN∩i_HK(K) =N∩K. FurthermoreNⁿ∩H = (N∩H)ⁿ andNⁿ∩K= (N∩K)ⁿ. This implies thatNⁿ∈∆₁ for alln≥1.

Theorem 4.1. LetG=ht, A;t⁻¹Ht=K, ϕibe an HNN extension whereH andK are finitely generated subgroups in the center ofA. SupposeAis subgroup separable andH 6=A6=K. Then Gisπc if and only if one of the following holds:

(a) H =K

(b) H *K, K*H and there exists a torsion free subgroupN /fHK such that ϕ(N∩H) =N∩K andN∩K, N∩H are isolated inN.

Proof. Suppose G is πc and suppose H 6= K. If H ⊂ K, then by Lemma 3.2, H = K, a contradiction. Therefore H 6= HK and similarly K 6= HK. So by Theorem 3.3, G1 = ht, HK;t⁻¹Ht = K, ϕi is πc and therefore residually finite.

Then by [11, Theorem], there exists a torsion free subgroup N /fHK such that ϕ(N∩H) =N∩K andN∩K, N∩H are isolated inN.

Conversely suppose H = K. Then G is π_c by Theorem 3.2. Next suppose H 6=K. Then H 6=HK 6=K. If there exists a torsion free subgroup N /_f HK such that ϕ(N∩H) = N ∩K and N ∩K, N∩H are isolated in N, then G₁ = ht, HK;t⁻¹Ht=K, ϕiis residually finite by [11, Theorem]. We will show thatG₁ isπ_c. By Theorem 3.1, it is sufficient to show that T

M∈∆1

M H=H, T

M∈∆1

M K=K and T

M∈∆1

Mhxi=hxi,∀x∈HK, where ∆₁={M /_fHK ; ϕ(M∩H) =M ∩K}.

First we show that T

N∈∆1

N H =H and T

N∈∆1

N K =K. SinceHandKare in the center ofA, the subgroupHKsatisfies the nontrivial identityW(x, y) =x⁻¹y⁻¹xy.

Therefore by [13, Theorem 3’], T

N∈∆1

N H=H and T

N∈∆1

N K =K.

Next we will show that T

M∈∆1

Mhxi=hxi,∀x∈HK. Lety ∈HK− hxi. Since HKis abelian and so subgroup separable, there existsM /_fHKsuch thaty /∈Mhxi.

By Lemma 4.1, there existsN ∈∆₁ such thatNⁿ ∈∆₁ for alln≥1. Since M is of finite index inHK, there exists a positive integer nsuch thatNⁿ ⊆M. Thus y /∈Nⁿhxiand so T

M∈∆1

Mhxi=hxi. Hence we have shown thatG₁ isπ_c. Now by

Theorem 3.3,Gisπ_c.

Corollary 4.1. Let G = ht, A;t⁻¹Ht = K, ϕi be an HNN extension where A is a polycyclic-by-finite group or a Fuchsian group. Suppose H and K are finitely generated subgroups in the center ofA andH 6=A6=K. ThenGisπ_c if and only if one of the following holds:

(a) H =K

(b) H *K, K*H and there exists a torsion free subgroupN /_fHK such that ϕ(N∩H) =N∩K andN∩K, N∩H are isolated inN.

Proof. Since polycyclic-by-finite groups and Fuchsian groups are subgroup separa-

ble, the corollary follows from Theorem 4.1.

Another application from Section 3 is the following result.

Corollary 4.2. Let G = ht, A;t⁻¹Ht = K, ϕi be an HNN extension where A is a polycyclic-by-finite group or a Fuchsian group. Suppose H and K are finitely generated subgroups in the center ofA such thatH∩K is finite. ThenGis πc. Proof. By Theorem 3.3, Gisπc if and only ifG1 =ht, HK;t⁻¹Ht=K, ϕi isπc. But by [15, Theorem 2],G1 isπc. HenceGisπc. References

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