Quasi-Potency and Cyclic subgroup separability

QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY JOS´ E BURILLO AND ARMANDO MARTINO Abstract. We study the profinite topology on discrete groups and in particular the property of cyclic subgroup separability. We investigate the class of quasi-potent, cyclic subgroup separable groups, producing many examples and showing how it behaves with respect to certain group constructions. Introduction The profinite topology on a discrete group Ghas, as a basis neighbourhood of 1, the finite index subgroups. We then say that Gis residually finite if the trivial subgroup is closed in this topology. Similarly, we say that G is subgroup separable if every finitely generated subgroup of Gis closed and cyclic subgroup separable if every cyclic subgroup is closed. Another way of defining separability is to say that a subgroup Hof Gis separable (or closed in the profinite topology) if for any g∈G−Hthere exists a finite quotient of Gin which the images of Hand gare disjoint. Thus it is clear that residual finiteness gives a solution to the word problem for Gand that any separable subgroup of Ghas solvable membership problem. However, subgroup separability is a very strong condition that is satisfied by very few classes of groups. For instance, it is well known that F2× F2has subgroups with unsolvable membership problem so they cannot be subgroup separable. There are even examples of 3-manifold groups and knot groups which are not subgroup separable (see [4] and [12]). Hence it seems reasonable to ask whether groups are separable with respect to a certain class of subgroups, and here we shall be concerned with the case of cyclic subgroups and cyclic separability. It turns out to be useful to impose the further restriction of quasi-potency (defined below) which allows one to construct finite quotients in some regular way. We thus define the class Cnto consist of those cyclic subgroup separable groups which are also quasi-potent. It is fairly easy to show that We would like to thank the Centre de Recerca Matem`atica for their support. 1 2 JOS´ E BURILLO AND ARMANDO MARTINO this class contains all finitely generated virtually free and virtually nilpotent groups. Moreover, this class is closed under taking amalgamated free products over virtually cyclic groups (Theorem 3.6). In fact, this class contains many examples and in general behaves well with respect to group constructions. For instance, it is fairly straightforward to show that the example shown in [4], of a 3-manifold group which is not subgroup separable, ha, b, t : [a, b], t−1atbi, lies in our class Cn. The class also contains all right-angled Artin groups (Theorem 3.8), all finitely generated residually free groups (Corollary 4.2) and all polycyclic groups (Corollary 5.5). It is also closed under taking direct sums and under certain extensions (Corollary 4.3 and Theorem 5.1). For instance, finitely generated free-by-free or surface-by-free groups are in the class Cn. One consequence of this is that, since there are examples of free-by-free groups with unsolvable conjugacy problem (see [10]), the class Cncontains groups which are not conjugacy separable. That is, they do not all have the property that conjugacy classes may be distinguished in finite quotients. 1. Notations and Definitions Throughout the paper, we will write H≤Gto indicate that His a subgroup of G,H£Gto indicate that His a normal subgroup of Gand Hch Gto indicate it is a characteristic subgroup. Additionally, we write H≤fG,H£fGand HchfGto denote that His, respectively a finite-index subgroup, a finite-index normal subgroup and a finite-index characteristic subgroup of G. The properties under study in this paper will be all strengthenings of the notion of residual finiteness. Recall that a group Gis called residually finite when for every element g6= 1 there exists a finite-index, normal subgroup H£fGsuch that g /∈H, i.e. ghas nontrivial image in G/H. One can specialize this definition by requiring certain values of the orders or indices for the objects involved in the definition of residual finiteness. One such specialisation is the condition of potency. For a group to be potent, there must be finite quotients where a given non-torsion element has arbitrary order. Definition 1.1. A group Gis called potent when for every element g6= 1 there exists a family of finite-index, normal subgroups Hn£fGsuch that hgi ∩ Hn=hgni, i.e. if ghas infinite order, the image of gin the quotient G/Hnhas order n. QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 3 This notion can be generalised by requiring that only some orders need to be realised, namely, those which are multiple of a fixed number. Definition 1.2. A group Gis called quasi-potent when for every element gthere exist a positive integer kg, which depends on g, and a family of finite-index, normal subgroups Hn£fGsuch that hgi ∩ Hn=hgnkgi, i.e. the image of gin the quotient G/Hnhas order nkg. The concept of quasi-potency for a group Gadmits the following equivalent characterisation, which is sometimes more useful: Remark 1.3. A group Gis quasi-potent if and only if the following two conditions hold: •Gis residually finite. •For each element gof infinite order, there exist a positive integer kg, which depends on g, and a family of finite quotients Qnof G such that ghas order nkgin Qn. Clearly, a potent group is quasi-potent, and both are residually finite. All these definitions can be modified by using characteristic subgroups instead of merely normal ones. Recall that a subgroup H≤Gis called characteristic if it is invariant under any automorphism of G. Clearly, every characteristic subgroup is normal. In this manner, we can define groups which are characteristically residually finite, characteristically potent and characteristically quasi-potent if the finite-index subgroups are required to be characteristic. Observe that if a group Gis finitely generated, then Gis characteristically residually finite if and only if it is residually finite, since one can construct a finite-index characteristic subgroup by taking the intersection of all automorphic images of a given finite-index normal subgroup. For arbitrary groups this is no longer true: The free abelian group of countably infinite rank is residually finite but it has no characteristic subgroups of finite index. The conditions of potency and quasi-potency and their characteristic counterparts are not related since any construction, such as the one above, changes the orders of elements in the quotients in complicated ways. Another strengthening is obtained when the quotient is required to be a p-group for a given prime number p. Hence, a group is called residually a finite p-group if, for every element g∈G, there exists a finite-index, normal subgroup Hsuch that g /∈H, and G/H is a p-group. Recall that the profinite topology of a group is the topology that has as a neighbourbood basis of the identity all subgroups of finite index. Namely, a basis for the topology is given by all cosets of all finite-index subgroups. 4 JOS´ E BURILLO AND ARMANDO MARTINO Residual finiteness admits an easy interpretation using the profinite topology: a group is residually finite if and only if its profinite topology is Hausdorff, or, equivalently, if the subgroup {1}is closed in the profinite topology. Analogously the notion of being residually a finite p-group can be read in terms of the pro-ptopology. The notions of potency and quasi-potency, since they involve orders of elements in the quotient, are not easily described in terms of the profinite and pro-ptopologies. The notion of residual finiteness has received considerable attention as have its generalisations to subgroup and cyclic subgroup separability, which is the subject of this paper. Definition 1.4. A group Gis called subgroup separable (also referred to as LERF), if every finitely generated subgroup is closed in the profinite topology. The group Gis called cyclic subgroup separable if every cyclic subgroup of Gis closed in the profinite topology. We note that a subgroup K≤Gis closed in the profinite topology if for every element g∈G−Kone can find a finite-index subgroup Hsuch that K⊂Hand g /∈H. Equivalently, gH ∩K=∅. Or, to put it yet in another way, elements lying outside a closed subgroup can be separated from the subgroup in some finite quotient. The next proposition is straightforward but we spell it out because of its usefulness. Proposition 1.5. Let Hbe a subgroup of a group G. •If G0≤fG, then His separable in Gif and only if H∩G0is separable in G0. •If H0≤fH, then if H0is separable in G, so is H. 2. The classes Cnand Cc Definition 2.1. The class Cnconsists of all groups which are quasi-potent and whose cyclic subgroups are separable. The class Cnadmits as a subclass the class Ccof groups which are characteristically quasi-potent and also whose cyclic subgroups are separable. Clearly, due to the quasi-potency condition, both classes Cnand Ccare subclasses of the class of residually finite groups. It is also clear that the class Cnis closed under taking subgroups, though the same is not apparent for Cc. It is our goal to prove that some known families of groups belong to the classes Cnand Cc. The main tool to accomplish this fact will be the following result: QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 5 Proposition 2.2. Let Gbe a finitely generated group which is residually a finite p-group for every prime number p. Then, Gis characteristically potent. Before we begin the proof, we will need some lemmas. Lemma 2.3. A finitely generated solvable group of finite exponent is finite. Proof. We argue by induction on the derived length of our solvable group, G. If Ghas derived length 1, then Gis abelian and the result is clear, so we shall assume that Ghas derived length nand the lemma holds for all solvable groups of derived length at most n−1. Then G/G0is a finitely generated abelian group of finite exponent and is therefore finite. Moreover, G0must also be finitely generated (since it has finite index in G), has finite exponent, and is itself a solvable group of derived length at most n−1. By the induction hypothesis, G0and hence Gis finite. tu Lemma 2.4. Let Gbe a group, Ka normal subgroup of Gand πthe projection from Gto G/K. Let H≤Gbe a subgroup which contains K. If H/K is a characteristic subgroup of G/K then His a normal subgroup of G. If Kis characteristic, then so is H. Lemma 2.5. Let Gbe a finite p-group and g∈Gan element of order pk≥p. Then there exists a characteristic subgroup Kof Gsuch that gK has order pk−1in G/K. Proof. As Ghas order pm, we proceed by induction on m, the case m= 1 being trivial. Gis a finite p-group and hence nilpotent. Thus its center Z(G) is a non-trivial characteristic abelian subgroup of G. The set of elements of order pin Z(G) forms a characteristic subgroup Hof Z(G), and hence also of G. Note that gH has order equal to either pk−1or pk in G/H. If gH has order pk−1, by taking K=Hwe are done. If gH has order pkin G/H, then, since G/H has order smaller than pm, the induction hypothesis applies to find a subgroup K/H in G/H such that the image of gH in G/K has order pk−1. Use Lemma 2.4 to finish the proof. tu Proof of Proposition 2.2. If K1, K2are finite index characteristic subgroups of Gthen so is K1∩K2. Moreover, the order of g(K1∩K2) in G/(K1∩K2) is the least common multiple of the orders of gK1and gK2, since G/(K1∩K2) is isomorphic to the image of the diagonal subgroup of G×Gin G/K1×G/K2. Hence, it is sufficient to prove that, given g∈Gthere exists a characteristic subgroup Ksuch that gK has order pmin G/K, for all primes pand for all positive integers m. During our proof we will find a family of characteristic subgroups of finite index of G, such that their intersection will be trivial. Consider the 6 JOS´ E BURILLO AND ARMANDO MARTINO descending central series of G, denoted by Γi(G), defined recursively by Γ0(G) = Gand Γi(G) = [Γi−1(G), G]. Observe that Γi(G) is a characteristic subgroup of G. Consider also the subgroup Gpm, for a given m, which is the subgroup generated by all the pm-th powers of the elements of G. Clearly, the subgroup Gpmis characteristic as well. Define the subgroup Gi= Γi(G)Gpi, and observe that, as the product of characteristic subgroups, it is also characteristic. We have that the subgroup Gihas finite index in G, as a consequence of Lemma 2.3. Moreover, G/Giis a finite p-group. Since Gis residually a finite p-group, and every map from gto a finite p-group factors through some G/Gi, we deduce that \ i Gi={1}. By Lemmas 2.4 and 2.5, it is sufficient to show that for a given g6= 1 in G, the orders of the element gGiin G/Giare unbounded. This can only fail if for some integer k,gpk∈Gifor all i. But since the intersection of the Giis trivial, this would imply that gpk= 1, which contradicts the fact that Gis torsion-free. tu From these results, one can now deduce that some groups which are residually a finite p-group for all pare in the classes Ccand Cn, namely, finitely generated free groups and nilpotent groups. Proposition 2.6. Let Fbe a finitely generated free group. Then (1) Fis residually a finite p-group for every p. (2) Cyclic subgroups in Fare separable Proof. These properties are both well known and in fact free groups are subgroup separable. See [7] or [11] for 1. Part 2 is a consequence of Marshall Hall’s Theorem, first proved in [6]. tu In the same way, we can deduce that finitely generated nilpotent groups are in these classes: Proposition 2.7. Let Gbe a finitely generated, torsion-free nilpotent group. Then (1) Gis residually a finite p-group for every p. (2) Cyclic subgroups of Gare separable. Proof. These results are elementary. In fact, finitely generated nilpotent groups are subgroup separable. See, for instance, [15], Theorem 4, page 19 and Exercise 11, page 18. tu The following result will allow to extend the result to groups that are virtually free or virtually nilpotent. QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 7 Lemma 2.8. Let Gbe a finitely generated group, which belongs to the class Cc, and such that any subgroup of finite index H≤fGalso belongs to the class Cc. Then, a group which is virtually G, that is, a group which has G as a subgroup of finite index, also belongs to Cc. Proof. Let Kbe a group such that G≤fK. Cyclic subgroup separability and residual finiteness of Kare immediate, so it only remains to show that K is characteristically quasi-potent. To do this we note that since Gis finitely generated and so is K, there are only finitely many subgroups in Kwith the same index as G. Their intersection will give a subgroup G0≤fGwhich is characteristic in K. By hypothesis, G0∈ Cc. Suppose now that we have a g∈Kof infinite order. Then gn∈G0, for some nand there exists an integer kgand a sequence of subgroups KnchfG0such that K0∩ hgni=hgnkgi. Since the Knare also characteristic of finite index in K, and the proof is complete. tu 3. Invariance under amalgams over virtually cyclic subgroups The goal for this section is to show that an amalgamated free product of groups in Cnover a virtually cyclic group is again in Cn. Some of this material has already appeared in [14] and [13], though we present it in order to give a slightly more general result, especially in view of our extension to certain HNN extensions in Theorem 3.7, and in order to present an elementary proof. We shall make use of the classical reduced form theorem for amalgamated free products. We shall follow the formulation in [5], Theorems 25 and 26. Theorem 3.1. Let G=G1∗HG2be an amalgamated free product. 1. Any g∈G−Hcan be written as x1. . . xn, where the xiare alternately from G1−Hand G2−H. This expression is called a reduced form for g. Such a product can never be in Hunless n= 1 and x1∈H. If g=x1∈H, then x1is the reduced form for g. 2. If we can also write g=y1. . . ym, with the yjbeing alternately from G1−Hand G2−H, then m=n,y1∈x1H,yn∈Hgnand yi∈HgiHfor all other i. 3. The number nis called the length of gand denoted by lG(g)or l(g)if G is understood; the length is well defined. We have that g∈G1∪G2if and only if l(g) = 1. 4. If g, h ∈Gthen l(gh)≥l(g)−l(h)and l(g−1) = l(g). An element g∈G1∗HG2with reduced form g=x1. . . xnis called cyclically reduced if either n= 1 or else if x1and xnbelong to different factors. The following lemma establishes some easy facts for cyclically reduced elements. 8 JOS´ E BURILLO AND ARMANDO MARTINO Lemma 3.2. Let g∈G1∗HG2. •If gis cyclically reduced then either l(g)=1or l(gk) = |k|l(g)for all integers k. In the former case, every power of glies in G1∪G2. •Every element of G1∗HG2is conjugate to a cyclically reduced element. The strategy to prove that the class Cnis closed under free amalgams over virtually cyclic groups is the following: take an element g∈G1∗HG2and write it in reduced form. By the residual finiteness of Giwe can find finiteindex subgroups Hi≤fGiwhich do not contain the elements involved in the reduced form of g. Then, consider the image of gin a possible amalgam of G1/H1and G2/H2. The only problem is that to construct this amalgam, one needs to make sure that H∩H1=H∩H2. This is the purpose of the following technical lemma. Lemma 3.3. Let G1, G2∈ Cn, let Hbe a virtually cyclic subgroup of both G1and G2and suppose that hxiis a finite index subgroup of H. Then, there exists an integer m, and sequences Ak£fG1,Bk£fG2such that Ak∩H=hxkmi=Bk∩H. Moreover, if N1, N2are normal, finite-index subgroups of G1and G2 respectively, then there exist subgroups, M1£fG1,M2£fG2, such that Mi≤Niand M1∩H=M2∩H. Proof. Let 1 = h1, . . . , hnbe coset representatives of hxiin H. As G1is cyclic subgroup separable, there exists a P1£fG1such that hiP1∩hxi=∅, or, equivalently, hi6∈ P1hxi, for all i6= 1. Clearly, this implies that P1∩H≤ hxi. By symmetry, we can also find a P2£fG2such that P2∩H≤ hxi. Also, as G1∈ Cn, due to quasi-potency, there exists an integer mx, and a sequence, Ak£fG1such that Ak∩ hxi=hxkmxi. As is customary, and to ease the notation, we will replace the sequence Akby a subsequence when needed, without changing the name Ak, and thus the number mxcan be replaced by any of its multiples. Combining the last two observations, we can assume, without loss of generality, that Ak∩ hxi ≤ P1∩H. Thus, by replacing each Akby Ak∩P1, again changing notation, we can assume that Ak∩H=hxkmxi. By symmetry, there exists an integer nxand a sequence, Bk£fG2such that Bk∩H=hxknxi. By taking further subsequences, we can then assume that mx=nx=m, by just replacing mxand nxby their least common multiple. This concludes the proof of the first assertion. To prove the second statement, suppose that N1∩ hxi=hxtiand N2∩ hxi=hxsi. Then we let M1=N1∩Ast and M2=N2∩Bst. Clearly, M1∩H=hxstmi=M2∩H.tu QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 9 Due to this last result, we shall use a particular construction: given an amalgamated free product G1∗HG2, we will take two subgroups N1£fG1 and N2£fG2with N1∩H=N2∩H. In this situation, we will denote by ΓN1,N2the amalgamated free product of the quotients, i.e., the product G1∗HG2where Gi=Gi/Niand H=H/(Ni∩H)∼ =HNi/Nifor both i= 1,2. Also, we have the natural map πN1,N2:G1∗HG2−→ ΓN1,N2 onto the amalgamated free product of the quotients. Observe that, in the case which interests us more, which is when the Gi=Gi/Niare finite, the group ΓN1,N2is virtually free, and hence, has many of the properties which we are interested in, and which will be suitable to pull back to G1∗HG2 with the map πN1,N2. The next result is a statement on the consistency of the definitions of length when we pass from Gto ΓN1,N2. Proposition 3.4. Let G1, G2∈ Cnand let Hbe virtually cyclic. Given 16=g∈G=G1∗HG2then there exists a ΓN1,N2with πN1,N2(g)6= 1 and lΓN1,N2(g) = lG(g). Moreover, for any ΓM1,M2, if Mi≤Nithen we similarly find that lΓM1,M2(g) = lG(g). Proof. The proof divides into two cases depending on whether lG(g) is bigger than one or equal to one. So suppose first that ghas length greater than one. Thus we can write it as a reduced product g=x1. . . xnwhere no xilies in H. Let S1={xi:xi∈G1}and S2={xi:xi∈G2}. Since H is separable in both G1and G2and since the sets Siare finite we can find A1£fG1and A2£fG2such that each Si∩HAiis empty. By Lemma 3.3 we can find N1£fG1,N2£fG2with Ni≤Aiand N1∩H=N2∩H. Clearly, each of Si∩HNiis empty. This means that if we look at the image of x1. . . xnin ΓN1,N2we get a reduced product of the same length. On the other hand, if lG(g) = 1 then assume that g∈G1. We can find an A1£fG1such that g6∈ A1. By applying Lemma 3.3 again, we get Ni£fGi such that πΓN1,N2(g) is different from 1 and has length one. The second assertion of this Lemma is obvious. tu The next result will imply separability for cyclic subgroups in G. Proposition 3.5. Let G1, G2∈ Cnand Hbe virtually cyclic. Consider ag∈G=G1∗HG2which is cyclically reduced and an h∈Gsuch that h6∈ hgi. Then we can find a ΓN1,N2such that πN1,N2(h)6∈ hπN1,N2(g)i. Proof. Again, the proof divides into two cases depending on the length of g. Suppose first that lG(g)>1. By repeated applications of Proposition 3.4 and Lemma 3.3 we can find a ΓN1,N2such that πN1,N2(grh)6= 1 for all integers rsuch that |r| ≤ lG(h). Moreover, we can ensure that 16 JOS´ E BURILLO AND ARMANDO MARTINO Proof. This is a corollary of Lemma 2.8, because a finite-index subgroup of a finitely generated residually free group is also finitely generated and residually free. tu Corollary 4.7. Surface groups are in Cc. Proof. Surface groups are residually free (see [2]). tu 5. Extensions Theorem 5.1. Consider a short exact sequence of groups, 1→N→G→K→1. Suppose that N∈ Ccis finitely generated and K∈ Cn. Further assume that for every N0≤fNwhich is normal in G, the quotient G/N0is residually finite. Then G∈ Cn. If, additionally, K∈ Ccis finitely generated and Nis a characteristic subgroup of Gthen G∈ Cc. Proof. The first thing to observe is that if N0is a characteristic subgroup of finite index in N, then there exists a finite index subgroup G0of Gsuch that G0∩N=N0. This is merely a consequence of the residual finiteness of G/N0, but immediately implies that Gis residually finite. Also note that we can always ensure that G0is normal in Gand if Nis characteristic in Gand Kis finitely generated we can further ensure that G0is characteristic in G. Quasi-potency is then fairly straightforward. Taking g∈Gwe need to find a sequence of finite quotients Qk, and a positive integer, mg, such that ghas order kmgin Qk. Since we already know that Gis residually finite, we can assume that ghas infinite order in G. If the image of galso has infinite order in K, then the quasi-potency of Kwill produce the sequence Qk. Note that if Nis characteristic in Gand K∈ Ccthen we can also ensure that these quotients are by characteristic subgroups. If the image of ghas finite order in K, we can replace gby a power and hence, without loss of generality, assume that g∈N. Hence we can find a sequence NkchfNsuch that ghas order kmgin N/Nkfor some positive integer mg. The remarks above ensure that we can find Gk£fGsuch that Gk∩N=Nkand this demonstrates quasi-potency. Further, if Kis finitely generated we can ensure that the Gkare characteristic. Thus Gis always quasi-potent, and is characteristically quasi-potent under the extra assumptions that Nis characteristic in Gand Kis finitely generated and in Cc. QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 17 Next we need to prove the separability of cyclic subgroups in G. So consider g, h ∈Gsuch that h6∈ hgi. Since hgiis closed in the profinite topology if some finite index subgroup of hgiis closed, we can replace gby a power. In particular we can assume that either g∈Nor the image of g has infinite order in K. Case 1: The image of ghas infinite order in K. Case 1 (i): grh∈N. Notice that since we are assuming gto have infinite order in K, the integer ris unique. Now, by hypothesis, grh6= 1 so there exists a finite index subgroup N0≤fNsuch that grh6∈ N0. As Nis finitely generated, we can assume that N0is characteristic in Nand hence normal in G. By hypothesis, G/N0is residually finite and hence there exists a finite index subgroup of G/N0which is isomorphic to a finite index subgroup of K. Thus G/N0is cyclic subgroup separable. By construction, we have that hN06∈ hgN0iand hence we can find a G0£fGsuch that h6∈ hgiG0. Case 1 (ii): h6∈ hgiN. Here the cyclic separability of Kensures that there exists a K0£fK such that hN 6∈ hgiNK0. Taking G0to be the preimage of K0in Gwe get that G0£fGand h6∈ hgiG0and we are done. Case 2: g∈N. Case 2 (i): h∈N. Then, by the cyclic separability of N, there exists a subgroup N0chfN such that h6∈ hgiN0. As above, we can find a G0£fGsuch that G0∩N= N0. Then we are done since it is clear that h6∈ hgiG0. Case 2 (ii): h6∈ N. As Kis residually finite, we can find a K0≤fKnot containing hN. Let G0be the preimage of K0in G. Then G0≤fGand it contains gbut not h. This completes the proof of cyclic separability and the proof of the theorem. tu 5.1. Applications. Theorem 5.1 provides criteria for when extensions of groups in Cn(or Cc) also belong to this class. However, the hypothesis that G/N0is residually finite is rather technical, so we will state several cases where it is satisfied. First, a definition. Given a short exact sequence 1→N→G→K→1, and a finite index subgroup G1of G, there is an induced short exact sequence, 1→N∩G1→G1→G1/(G1∩N)→1. We shall say that the former short exact sequence virtually splits if there exists some G1≤fGsuch that the latter sequence splits. 18 JOS´ E BURILLO AND ARMANDO MARTINO Proposition 5.2. If the short exact sequence in Theorem 5.1 virtually splits, then G/N0is always residually finite. That is, the technical hypothesis is always satisfied. Proof. Let G1≤fGbe a subgroup for which the induced sequence splits. Then, G1=N1oK1, where N1= (N∩G1) and K1∼ =G1/(G1∩N). Note that K1is isomorphic to G1N/N which is a subgroup of Kand hence residually finite. If N0≤fNand is normal in Gthen, G1/(G1∩N0) = G1/(N1∩N0)∼ =N1/(N1∩N0)oK1, since K1∩N1= 1. Thus G1/(G1∩N0) must be residually finite, since it has K1as a subgroup of finite index. However, G1/(G1∩N0)∼ =G1N0/N0≤fG/N0, and thus G/N0has a subgroup isomorphic to G1/(G1∩N0) as a subgroup of finite index so it is also residually finite.2 As an application we get the following, though it is clear for elementary reasons. Corollary 5.3. The hypothesis of Theorem 5.1 is satisfied if Kis virtually free. Another situation where we get the same behaviour is with surface groups, Proposition 5.4. The hypothesis of Theorem 5.1 is satisfied if Kis virtually a surface group. Proof. By [8], Exercise 4.7, page 60, any group Gwhich fits into a short exact sequence, 1→F→G→Γ→1, where Fis finite and Γ is a hyperbolic surface group is residually finite. The case of a torus is dealt with below, Theorem 5.10. 2 5.2. Polycyclic. We can apply Theorem 5.1 to polycyclic groups. By [15], Theorem 4, page 35, given a polycyclic group P, there exists a finite index subgroup P0which is nilpotent-by-abelian. In particular, it has a nilpotent derived subgroup. Moreover, any subgroup of P0also has a nilpotent derived subgroup. Thus we can consider the short exact sequence 1→[P0, P0]→P0→P0/[P0, P0]→1, where both the group on the left and the group on the right are in Cc and moreover, the group [P0, P0] is characteristic in P0. Since P0/[P0, P0] is abelian, we immediately see that the hypotheses for Theorem 5.1 are satisfied and hence that P0∈ Cc. Moreover, the same argument shows that every subgroup of P0is also in Cc. Thus, by Lemma 2.8, Pis also in Cc. QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 19 Theorem 5.5. All polycyclic subgroups are in Cc. We shall now show that the hypotheses of Theorem 5.1 are satisfied whenever the group Kis a polycyclic group. Recall that the ascending chain condition for a group Gis the property that all ascending chains of subgroups of Gare eventually stable. Equivalently, it states that all subgroups of Gare finitely generated. Recall also that polycyclic groups are precisely those solvable groups which satisfy the ascending chain condition. Thus the free abelian group of countably rank is solvable but not polycyclic. For a finitely generated example, take the free metabelian group F/[F0, F0] where Fis the free (non-abelian) group of rank 2 and F0the derived subgroup. It is well known that F0is a free group of countably infinite rank and thus that F0/[F0, F0] is free abelian of countably infinite rank. Hence F/[F0, F0] is finitely generated, solvable but not polycyclic. Proposition 5.6. Virtually polycyclic groups are subgroup separable. Proof. [15], Exercise 11, page 18. tu We will need the following result about polycyclic groups. Its proof can be found in [3]. Lemma 5.7. Let Gbe a finitely generated group with a finite, normal subgroup Nsuch that G/N is abelian. Then Gis virtually abelian. We will also need the following result about the ascending chain condition: Lemma 5.8. Consider a short exact sequence of groups, 1→A→B→C→1. If Aand Care finitely generated, then so is B. Moreover, if both Aand C satisfy the ascending chain condition, then so does B. Proof. The first claim is clear. For the second claim consider a subgroup H≤A. Then we get a short exact sequence, 1→H∩A→H→H/(H∩A)→1. The second assertion of the lemma now follows once we recall the second isomorphism theorem, H/(H∩A)∼ =HA/A ≤B/A ∼ =C. tu We finally return to polycyclic groups to establish our final results:. 20 JOS´ E BURILLO AND ARMANDO MARTINO Proposition 5.9. Let Gbe a group with a finite normal subgroup, N, such that G/N is polycyclic. Then Ghas a finite-index polycyclic subgroup. Proof. By Lemma 5.8, all subgroups of Gare finitely generated. We argue by induction on the derived length of G/N. For derived length equal one we appeal to Lemma 5.7. Otherwise, we can apply an induction hypothesis to deduce that G0, the derived subgroup of G, contains a finite-index polycyclic subgroup H1. Here it is where we are using that Gsatisfies the ascending chain condition. Without loss of generality, we can assume that H1is a normal subgroup of G. Hence G/H1is a finitely generated group, whose quotient by G0/H1 is abelian. Thus, again by Lemma 5.7, Gcontains a finite index subgroup H0such that H0/H1is abelian. Since H0is finitely generated and H1is polycyclic, H0must also be polycyclic. tu Corollary 5.10. The hypotheses of Theorem 5.1 are satisfied whenever N∈ Ccand Kis polycyclic-by-finite. References 1. R. B. J. T. Allenby and David Doniz, A free product of finitely generated nilpotent groups amalgamating a cycle that is not subgroup separable, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1003–1005. MR MR1350930 (97a:20049) 2. Gilbert Baumslag, On generalised free products, Math. Z. 78 (1962), 423–438. MR MR0140562 (25 #3980) 3. Martin R. Bridson and Andr Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer–Verlag, Berlin, 1999. MR MR1744486 (2000k:53038) 4. R. G. Burns, A. Karrass, and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Austral. Math. Soc. 36 (1987), no. 1, 153–160. MR MR897431 (88g:20057) 5. Daniel E. Cohen, Combinatorial group theory: a topological approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR MR1020297 (91d:20001) 6. Marshall Hall, Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421–432. MR MR0032642 (11,322e) 7. Kenkiti Iwasawa, Einige S¨atze ¨uber freie Gruppen, Proc. Imp. Acad. Tokyo 19 (1943), 272–274. MR MR0014089 (7,239d) 8. Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkh¨auser Boston Inc., Boston, MA, 2001. MR MR1792613 (2002m:57018) 9. Goansu Kim and C. Y. Tang, Cyclic subgroup separability of HNN-extensions with cyclic associated subgroups, Canad. Math. Bull. 42 (1999), no. 3, 335–343. MR MR1703693 (2000f:20043) 10. Ch. F. Miller, III, Decision problems in algebraic classes of groups (a survey), Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. QUASI-POTENCY AND CYCLIC SUBGROUP SEPARABILITY 21 California, Irvine, Calif. 1969; dedicated to Hanna Neumann), North-Holland, Amsterdam, 1973, pp. 507–523. Studies in Logic and the Foundations of Math., Vol. 71. MR MR0399270 (53 #3121) 11. Lee Neuwirth, An alternative proof of a theorem of Iwasawa on free groups, Proc. Cambridge Philos. Soc. 57 (1961), 895–896. MR MR0130911 (24 #A765) 12. Graham A. Niblo and Daniel T. Wise, Subgroup separability, knot groups and graph manifolds, Proc. Amer. Math. Soc. 129 (2001), no. 3, 685–693. MR MR1707529 (2001f:20075) 13. L. Ribes, D. Segal, and P. A. Zalesskii, Conjugacy separability and free products of groups with cyclic amalgamation, J. London Math. Soc. (2) 57 (1998), no. 3, 609–628. MR MR1659845 (99m:20053) 14. Luis Ribes and Pavel A. Zalesskii, Conjugacy separability of amalgamated free products of groups, J. Algebra 179 (1996), no. 3, 751–774. MR MR1371742 (97a:20046) 15. Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR MR713786 (85h:20003) Escola Polit` ecnica Superior de Castelldefels, UPC, Avda del Canal Ol´ ımpic s/n, 08860 Castelldefels, Barcelona, Spain) E-mail address:[email protected] Armando Martino, Centre de Recerca Matematica, Bellaterra, Spain E-mail address:[email protected]