Some groups which are not poly CF

In this section we present some results which require very little further effort to prove, since we may use Corollary 2.7 and the proofs of Propositions 1.29, 1.30 and 1.31 (Theorems 12, 13 and 16 in [9]).

Proposition 3.3. A finitely generated nilpotent group is poly-CF if and only if it is virtually abelian.

Proof. A finitely generated virtually abelian group is poly-CF by Observation 3.1. The rest of the proof is copied almost word-for-word from the proof of Theorem 12 in [9].

Assume thatGis a finitely generated nilpotent but not virtually abelian group. By Corollary 1.5,Ghas a torsion-free nilpotent subgroup of finite index, which cannot be virtually abelian sinceGis not virtually abelian. Furthermore, every non-abelian torsion-free nilpotent group has a subgroup isomorphic to the Heisenberg group

H=hA, B, C|[A, B] =C,[A, C] = [B, C] = 1i.

To see this letA be a non-central element of the second term of the upper central series ofGand let B be some element not commuting withA.

Since the poly-CF groups are closed under taking finitely generated subgroups, it suffices to show that H is not poly-CF. Since [Am, Bm] = Cm2 holds in H for allm ≥0, the commutative image of the intersection of (A−1)∗(B−1)∗A∗B∗(C−1)∗ with W(H,{A, B, C}) satisfies the condition of Proposition 1.28. (Given k ∈ _N, considera= (m, m, m, m) withm≥4k.) Thus H, and henceG, is not poly-CF by Corollary 2.7.

Note that this shows that the class of poly-CF groups is not closed under taking semidirect products, even with context-free top group, since the Heisenberg group is a semidirect productZ2o Z.

Next we will give an analogue of Proposition 1.30 for poly-CF groups, but first we need the promised completion of the proof of Proposition 1.30. This uses the following result, known as the Kurosh Subgroup Theorem - a special case of the Grushko-Neumann Theorem for free products. A proof can be found in [16, III.3.6].

Proposition 3.4. Let Gbe the free product of groupsGi,i∈I, whereI is an index

with groups that are conjugates of subgroups of the free factorsGi of G.

Proposition 3.5. For m ∈ _Z\ {0}, the Baumslag-Solitar group BS(m,±m) is virtually a direct product of two free groups and is thus both coCF and 2-CF.

Proof. First letG=BS(m, m) =

x, y|y−1_xm_y=xm

. Thenxm_{∈Z(G) and}

G/hxmi=hx, y|xmi=Cm∗Z.

LetH/hxmibe the normal closure in G/hxmiof hyi. Then

|G/hxmi:H/hxmi |=m

and hence|G:H|=m. SinceH/hxmi does not intersect any conjugate of Cm, by the Kurosh Subgroup Theorem (Proposition 3.4),H/hxmi is the free product of a free group with conjugates ofZ, and is thus free. Also, H∼=H/hxmi × hxmi, since xm∈Z(G). ThusGis virtually a direct product of two free groups.

Now letG=BS(m,−m) =x, y|y−1xmy=x−m.LetK be the normal closure in

Gof

x, y2

, which has index 2 in G. Settinga=x,b=y−1x−1y and c=y2 gives

K =ha, b, c|am =bm,[am, c]i,

witham∈Z(K), since (am)b = (bm)b =bm=am. Now take

H:=K/hami=ha, b, c|am =bm= 1i=Cm∗Cm∗Z.

Let φ be the homomorphism from H to Cm ×Cm given by mapping a onto a generator of the first Cm and b onto a generator of the second Cm, and c onto the identity. Then the intersection of kerφwith every conjugate ofhaiandhbiis trivial. Thus by the Kurosh Subgroup Theorem, kerφis a free product of a free group and conjugates of Z, and is hence itself free. Also,|H: kerφ|=|Cm×Cm|=m2. Let

K1 be the preimage of kerφ in K. Since kerφ is free and hami ∈ Z(H), K1 is

since kerφhas finite index inH=K/hami. ThusGis virtually a direct product of two free groups.

Hence G is 2-CF by Observation 3.1, and coCF by the fact that the coCF groups are closed under taking finite direct products [9, Proposition 6].

We can now determine which Baumslag-Solitar groups are poly-CF.

Proposition 3.6. The Baumslag-Solitar group BS(m, n) is poly-CF if and only if

m=±n.

Proof. LetG=BS(m, n) =x, y|y−1xmy=xn, wherem, n∈Z\ {0}. By Propo-

sition 3.5, if m=±m, then Gis poly-CF. The rest of the proof is copied from the proof of Theorem 13 in [9].

We deal here with the case 0 < m < n, the other cases being similar. Let L be the commutative image of W(G)∩(y−1)∗(x−1)∗y∗x∗. Then (k, mk, k, nk) ∈ L for all l ∈ _N, and nk is the only value of x for which (k, mk, k, x) ∈ L. Moreover, since n > m, for every given l there exists k with l(2k+mk) ≤ nk. This simply means that Lsatisfies the hypothesis of Proposition 1.28, so W(G) is not poly-CF

by Corollary 2.7.

Proposition 1.31 (Theorem 16 in [9]) also has its analogue for poly-CF groups. We do not give the proof in full, since the proof of Theorem 16 in [9] is quite long.

Proposition 3.7. If G is a finitely generated polycyclic group, then G is poly-CF

if and only if Gis virtually abelian.

Proof. The proof of Theorem 16 in [9] shows that ifG is a finitely generated poly- cyclic group which is not virtually abelian, then W(G) can be intersected with a regular language to give a sublanguage satisfying the hypothesis of Proposition 1.28, and so W(G) is neither coCF nor poly-CF by Corollary 2.7.