Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

HAL Id: hal-00184740

https://hal.archives-ouvertes.fr/hal-00184740

Submitted on 31 Oct 2007

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

Daciberg Gonçalves, John Guaschi

To cite this version:

Daciberg Gonçalves, John Guaschi. Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane. Journal of Group Theory, De Gruyter, 2010, 13 (2), pp.277-294.

�10.1515/JGT.2009.040�. �hal-00184740�

hal-00184740, version 1 - 31 Oct 2007

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

DACIBERG LIMA GONC ¸ ALVES Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo,

CEP: 05311-970 - S˜ao Paulo - SP - Brazil.

e-mail: [email protected] JOHN GUASCHI

Laboratoire de Math´ematiques Nicolas Oresme UMR CNRS 6139, Universit´e de Caen BP 5186,

14032 Caen Cedex, France.

e-mail: [email protected] 31st October 2007

Abstract

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups Pn^pR_P²q of the projective plane. The maximal finite subgroups of Pn^pR_P²q are isomor- phic to the quaternion group of order 8 if n 3, and to Z_{4 if n} ¥ 4. Further, for all n ^¥3, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of Pn^pR_P²q:Z_,Z₂Zand the amalgamated productZ₄Z₂Z_4.

1 Introduction

The braid groups B_n of the plane were introduced by E. Artin in 1925 [A1, A2]. Braid groups of surfaces were studied by Zariski [Z]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition [FoN]. Let M be a compact, connected surface, and letn ^PN. We denote the set of all orderedn-tuples of distinct points ofM, known as then^th configuration space of M, by:

F_n^pM^q

!

pp₁, . . . ,p_n^q

p_i ^P Mand p_i p_j ifi j

)

.

Configuration spaces play an important r ˆole in several branches of mathematics and have been extensively studied, see [CG, FH] for example.

2000 AMS Subject Classification: 20F36 (primary)

The symmetric groupS_nonnletters acts freely onF_n^pM^qby permuting coordinates.

The corresponding quotient will be denoted byD_n^pM^q. Then^thpure braid group P_n^pM^q (respectively then^thbraid group B_n^pM^q) is defined to be the fundamental group ofF_n^pM^q (respectively of D_n^pM^q).

Together with the 2-sphereS², the braid groups of the real projective planeR_P²_are of particular interest, notably because they have non-trivial centre [VB, GG1], and tor- sion elements [VB, Mu]. Indeed, Van Buskirk showed that among the braid groups of compact, connected surfaces, B_n^pS²q and B_n^pR_P²q are the only ones to have tor- sion [VB]. Let us recall briefly some of the properties ofB_n^pR_P²q[GG1, Mu, VB].

IfD² „R_P²is a topological disc, there is a group homomorphismι: B_n ^ÝÑB_n^pR_P²q

induced by the inclusion. If β ^P B_n then we shall denote its image ι^pβ^q simply by β. A presentation of B_n^pR_P²q was given in [VB], and of P_n^pR_P²q in [GG3]. The first two braid groups of R_P² are finite: B₁^pR_P²q P₁^pR_P²q Z_{2, P2}pR_P²qis isomorphic to the quaternion group Q₈ of order 8 and B₂^pR_P²q is isomorphic to the generalised quaternion group of order 16. For n ^¥ 3, B_n^pR_P²q is infinite. The pure braid group P₃^pR_P²qis isomorphic to a semi-direct product of a free group of rank 2 byQ₈[VB]; an explicit action was given in [GG1] (see also the proof of Proposition 18).

The so-called ‘full twist’ braid of B_n^pR_P²q is defined by ∆_n pσ₁σ_n1^qn. For n ^¥ 2, ∆_n is the unique element of B_n^pR_P²q of order 2 [GG1], and it generates the centre of B_n^pR_P²q [Mu]. The finite order elements of B_n^pR_P²q were characterised by Murasugi [Mu] (see Theorem 6, Section 3), however their orders are not clear, even for elements of P_n^pR_P²q. In [GG1], we proved that for n ^¥ 2, the torsion of P_n^pR_P²qis 2 and 4, and that ofB_n^pR_P²qis equal to the divisors of 4nand 4^pn1^q.

The classification of the finite subgroups of B_n^pS²q and B_n^pR_P²q is an interesting problem, and helps us to better understand their group structure. In the case ofS²_{, this} was undertaken in [GG2, GG4]. It is natural to ask which finite groups are realised as subgroups ofB_n^pR_P²q. As for B_n^pS²q, one common property of such subgroups is that they are finite periodic groups of cohomological period 2 or 4. Indeed, by Proposition 6 of [GG1], the universal covering X of F_n^pR_P²q is a finite-dimensional complex which has the homotopy type of S³. Thus any finite subgroup of B_n^pR_P²q acts freely on X, and so has period 2 or 4 by Proposition 10.2, Section 10, Chapter VII of [Br]. Since∆_n is the unique element of order 2 of B_n^pR_P²q, and it generates the centreZ^pB_n^pR_P²qq, the Milnor property must be satisfied for any finite subgroup ofB_n^pR_P²q.

In Section 2, we start by determining the maximal finite subgroups of Pn^pR_P²q: PROPOSITION 1.Up to isomorphism, the maximal finite subgroups of P_n^pR_P²qare:

(a) Z_{2if n}1.

(b) Q₈if n2, 3.

(c) Z_{4if n}¥4.

In Section 3, we simplify Murasugi’s characterisation of the torsion elements of B_n^pR_P²q (see Proposition 7), which enables us to show that within B_n^pR_P²q, there are two conjugacy classes of subgroups isomorphic to Z_{4 lying in Pn}pR_P²q (see Proposi- tion 9).

The rest of the paper is devoted to determining the infinite virtually cyclic sub- groups of P_n^pR_P²q(recall that a group is said to bevirtually cyclicif it has a cyclic sub- group of finite index). This was initially motivated by a question of Stratos Prassidis concerning the determination of the algebraic K-theory of the braid groups of S² _and

R_P². It has been shown recently that the full and pure braid groups of these surfaces satisfy the Fibered Isomorphism Conjecture of T. Farrell and L. Jones [BJPL, JPML1, JPML2]. This implies that the algebraic K-theory groups of their group rings may be computed by means of the algebraicK-theory groups of their virtually cyclic subgroups via the assembly maps. More information on these topics may be found in [BLR, FJ, JP].

As well as helping us to better understand these braid groups, this provides us with ad- ditional reasons to find their virtually cyclic subgroups.

In Section 4, we recall the criterion due to Wall of an infinite virtually cyclic group G as one which has a finite normal subgroup F such thatG^{F is isomorphic toZ _{or to} the free productZ₂Z₂ (this being the case, we shall say that G is of Type I orType II respectively). Wall’s result enables us to establish a list of the possible infinite virtually cyclic subgroups of a given groupG, if one knows its finite subgroups (which is the case for P_n^pR_P²q). The real difficulty lies in deciding whether the groups belonging to this list are effectively realised as subgroups ofG. In Lemma 16 we give a useful criterion to decide whether a group is of Type II.

In the case of P_n^pS²q, since there are only two finite subgroups forn ^¥ 3, the trivial group and that generated by the full twist (which are the elements of the centre of P_n^pS²q), it is then easy to see that its infinite virtually cyclic subgroups are isomorphic toZ_{or to}Z₂Z_.

Since the structures of the finite subgroups of P₃^pR_P²qandP_n^pR_P²qdiffer, we sepa- rate the discussion of their virtually cyclic subgroups. However, it turns out that up to isomorphism, they have the same infinite virtually cyclic subgroups:

THEOREM2.Let n^¥3. Up to isomorphism, the infinite virtually cyclic subgroups of P_n^pR_P²q

areZ_,Z₂Z_andZ₄Z₂Z_4.

Forn3, the result will be proved in Theorem 17 (see Section 5.1), while forn ^¥4, it will be proved in Theorem 19 (see Section 5.2).

As a immediate corollary of Proposition 1 and Theorem 2, we obtain the classifica- tion of the virtually cyclic subgroups ofP_n^pR_P²q:

COROLLARY 3.Let n ^P N. Up to isomorphism, the virtually cyclic subgroups of P_n^pR_P²q

are:

(a) The trivial group^te^uandZ_{2if n}1.

(b) ^te^u,Z_2,Z_{4andQ8if n}2.

(c) ^te^u,Z_2,Z_4,Q8,Z_,Z₂Z_andZ₄Z₂ Z_{4if n}3.

(d) ^te^u,Z_2,Z_4,Z_,Z₂Z_andZ₄Z₂Z_{4if n}¥4.

One of the key results needed in the proof of Theorem 2 is that P_n^pR_P²q has no subgroup isomorphic toZ₄Z. This fact allows us to eliminate several possible Type I and Type II subgroups, and has the following interesting corollary which we prove at the end of Section 5.2:

COROLLARY 4.Let n ^¥2, and let x ^P P_n^pR_P²qbe an element of order4. Then its centraliser Z_PnpR_P2q

px^qin P_n^pR_P²qis equal to^xx^y.

The study of the finite subgroups ofB_n^pR_P²qand of the infinite virtually cyclic sub- groups ofB_n^pS²qand B_n^pR_P²qis the subject of work in progress.

Acknowledgements

This work took place during the visits of the second author to the Departmento de Matem´atica do IME-Universidade de S˜ao Paulo during the periods 10^th – 27^thAugust 2007 and 4^th – 23^rd October 2007, and of the visit of the first author to the Laboratoire de Math´ematiques Emile Picard, Universit´e Paul Sabatier, Toulouse, during the period 27^th August – 17^th September 2007. It was supported by the international Cooperation USP/Cofecub project number 105/06, by an ‘Aide Ponctuelle de Coop´eration’ from the Universit´e Paul Sabatier, by the Pr ´o-Reitoria de Pesquisa Projeto 1, and by the Projeto Tem´atico FAPESP no. 2004/10229-6 ‘Topologia Alg´ebrica, Geometrica e Diferencial’.

The authors would like to thank Stratos Prassidis for having brought the question of the virtually cyclic subgroups to their attention.

2 Finite subgroups of P

R _P

In this section, we characterise the finite subgroups of P_n^pR_P^2q by proving Proposi- tion 1. In Theorem 4, Corollary 19 and Proposition 23 respectively of [GG1], we ob- tained the following results:

THEOREM 5 ([GG1]).Let n^¥2. Then:

(a) B_n^pR_P²qhas an element of orderℓif and only ifℓdivides either4n or4^pn1^q. (b) the (non-trivial) torsion of P_n^pR_P²qis precisely2and4.

(c) the full twist∆_n is the unique element of B_n^pR_P²qof order2.

It was shown in [Mu] that∆_n generates the centre ofB_n^pR_P²q. Using the projection P_{n 1}^pR_P²q ÝÑ P_n^pR_P²q given by forgetting the last string, and induction, one may check that∆_n also generates the centre ofP_n^pR_P²q.

It follows from Theorem 5 that the maximal (finite) cyclic subgroups ofB_n^pR_P²qare isomorphic toZ_{4n or}Z₄

pn1^q, and that those ofP_n^pR_P²qare isomorphic toZ_4.

Proof of Proposition 1. Since P₁^pR_P²q Z_{2 and P2}pR_P²q Q₈, the result follows easily forn1, 2. So suppose thatn^¥3. Consider the Fadell-Neuwirth short exact sequence:

1^ÝÑP_n2^pR_P²ztx₁,x₂^uqÝÑP_n^pR_P²q

p

ÝÑP₂^pR_P²qÝÑ1,

wherepcorresponds geometrically to forgetting the lastn2 strings. LetHbe a finite subgroup ofP_n^pR_P²q. Thenp|_H is injective. To see this, suppose thatx,y ^P Hare such that p^px^q p^py^q. Then xy^{1 P} H^XKer^pp^q. But Ker^pp^q is torsion free, so x y, which proves the claim. In particular,|H|^¤8, and since the torsion ofP_n^pR_P²qis 2 and 4, and P_n^pR_P²qhas a unique element of order 2, it follows that H is isomorphic to one of^te^u,Z_{2(so is}x∆_ny),Z_{4orQ8. Ifn} 3 then the above short exact sequence splits [VB], and so P₃^pR_P²q F₂Q₈. Hence the finite subgroups of P₃^pR_P²q are those of Q₈. Now let n ^¥ 4. Then Q₈ is not realised as a subgroup of P_n^pR_P²q. For suppose that H   P_n^pR_P²q, where H Q₈. From above, it follows that p|_H is an isomorphism, so p admits a section. But this would contradict Theorem 3 of [GG1]. The result follows from Theorem 5(b).

3 Murasugi’s characterisation of the finite order elements of B

R _P

In this section, our aim is to reorganise the classification of Murasugi [Mu] of the finite order elements ofB_n^pR_P²qin a form that shall be more convenient for our purposes. As well as being useful in its own right, we shall make use of this reorganisation to prove that there are precisely two conjugacy classes in B_n^pR_P²qof subgroups of pure braids of order 4. This shall be applied in the classification of the virtually cyclic subgroups of P_n^pR_P²q, notably to show that P_n^pR_P²qhas no subgroup isomorphic to Z₄Z _(see Propositions 18 and 22).

Up to conjugacy, the finite order elements of B_n^pR_P²q may be characterised as fol- lows:

THEOREM 6 ([MU]).Every element of B_n^pR_P²qof finite order is conjugate to one of the fol- lowing elements:

A₁^pn,r,s,q^qpρrσ_r1σ₁^qspσ_{r 1}σ_n1^qq A₂^pn,r,s,q^qpρrσ_r1σ₁^qspσ_{r 1}σ_n1σ_{r 1}^qq,

where ^pnr,q^{q p}2r,s^q in the first case, and ^pnr1,q^{q p}2r,s^q in the second. The relationis defined by:

pa,b^qpc,d^qif there exists^pm,k^qp0, 0^qsuch that m^pa,b^qk^pc,d^q. This characterisation may be simplified as follows:

PROPOSITION 7.Let n ^¥ 2. Then every finite order element of B_n^pR_P²q is conjugate to a power of some A_i^pn,r, 2r^{l,p^{l^q, i 1, 2, where

p

$

&

%

nr if i1 nr1 if i2, and l gcd^pp, 2r^q.

REMARK 8. The characterisation of the finite order elements of B_n^pR_P²qthus appears more transparent than that given in Theorem 6, in the sense that these elements may be determined directly fromn and r (without involving the integerss,q, nor the relation

).

Proof of Proposition 7. From Theorem 6, every finite order element of B_n^pR_P²qis conju- gate to some A_i^pn,r,s,q^q, where m^pp,q^q k^p2r,s^q. Thus every finite order element of B_n^pR_P²q is conjugate to a power of some A_i^pn,r,s^{λ,q^{λ^q, where λ gcd^ps,q^q. We consider the following cases separately:

(a) If r 0 then m s 0 and k 0. Then A_i^pn,r,s^{λ,q^{λ^q A_i^pn, 0, 0, 1^q A_i^pn,r, 2r^{l,p^{l^qas required.

(b) If eitheri 2 andr n1, or i 1 andr nthen p 0. Thusk 0 and m 0, soq 0. HenceA_i^pn,r,s^{λ,q^{λ^q A_i^pn,r, 1, 0^q A_i^pn,r, 2r^{l,p^{l^qas required.

(c) Suppose that eitheri ^Pt1, 2^uand 1^¤r^¤n2, ori1 andrn1, sop 0. Since mp 2rk,mqksand^pm,k^qp0, 0^q, it follows thatm,k0,m^{k s^{q 2r^{p(ifs0 thenq0, but thenA_i^pn,r,s,q^qwould be trivial), and sos¹p2rq¹, wheres¹ s^{λand q¹ q^{λare coprime. We thus obtain 2r ls¹and p lq¹, so ^p2r,p_l ^{q ps,q}_λ^q. Hence every finite order element ofB_n^pR_P²qis conjugate to a power of someA_i^pn,r, 2r^{l,p^{l^q.

In Proposition 26 of [GG1], we proved that the following elements ofB_n^pR_P²q: aσ_n¹

1σ₁¹ρ₁ bσ_n¹

2σ₁¹ρ₁

are of order 4nand 4^pn1^qrespectively. By Remark 27 of [GG1], we have

$

&

%

α aⁿ ρ_nρ₁

βbⁿ¹ ρ_n1ρ₁. (1) It is clear thatαandβare pure braids of order 4. Using Proposition 7, we now show that any element inPn^pR_P²qof order 4 is conjugate inBn^pR_P²qto one of these two elements (or their inverses):

PROPOSITION9.Let n^¥2. Then every element of order4of P_n^pR_P²qis conjugate in B_n^pR_P²q

to one ofα andβor their inverses.

REMARK 10. Since the Abelianisation ofB_n^pR_P²qis isomorphic to Z₂Z₂ (cf. the pre- sentation of B_n^pR_P²q given in [VB]), and the generator σ_i, 1 ^¤ i ^¤ n1 (resp. ρ_j, 1 ^¤ j ^¤ n) is sent to the generator of the first (resp. second) Z_{2-factor,α and βare not} conjugate inB_n^pR_P²q.

Proof of Proposition 9. For givennandr, define ω

$

&

%

σ_{r 1}σ_n1 ifi 1 σ_{r 1}σ_n1σ_{r 1} ifi 2.

Up to conjugacy and powers, the finite order elements ofB_n^pR_P²qappear in one of the three cases given in the proof of Proposition 7. Among these elements, we search for pure braids of order 4.

(a) We have that A_i^pn, 0, 0, 1^q ω. Ifi 1 then ω σ₁σ_n1is of order 2n, and its permutation is^p1,n, , 2^q. The smallestj^¥1 for whichω^jbecomes pure is thusjn, but thenωⁿ ∆_n. Ifi 2 thenω σ₁σ_n1σ₁is of order 2^pn1^q, its permutation is

p1,n, , 3^q, and ωⁿ¹ ∆_n. The smallest j ^¥ 1 for whichω^j becomes pure is indeed jn1. In either case, the powers ofω yield no pure braids of order 4.

(b) Suppose that either i 2 and r n1, or i 1 andr n. Using the relation ρ_{i 1} σ_i¹ρ_iσ_i¹, 1^¤i^¤n1 inB_n^pR_P²q(see [VB, Mu, GG1]), we have that

A_i^pn,r, 1, 0^qρ_rσ_r1σ₁

$

&

%

b ifrn1 a ifrn.

Studying the permutations of a and b, we observe that the smallest power of each of these elements which is a pure braid isbⁿ¹βandaⁿ α(which are of order 4).

(c) If i 1 and r n1 then we have A₁^pn,r, 2r^{l,p^{l^q A₁^pn,n1, 2^pn1^q, 1^q

pρ_n1σn2σ₁^q2pn1q b^2pn1q ∆_n which is of order 2. So we may suppose that 1 ^¤ r ^¤ n2. Letξ ρ_rσ_r1σ₁andy_i A_i^pn,r, 2r^{l,p^{l^q ξ^2r{lω^p{l. By equation (5.1) at the bottom of page 79 of [Mu],y^l_i ξ^2rω^p ∆_n, so the order ofy_idivides 2l. Now the permutation ofωis ap-cycle, and sinceldividesp, the permutation ofω^p{lis a product of p^{l disjointl-cycles. Thus the smallest j ^¥ 1 for which^pω^p{lqjbecomes a pure braid isj l. But y^l_i is the full twist, and so the only pure braids among the powers ofy_iare

∆_n and the identity. In particular, no power ofy_ican yield a pure braid of order 4.

We conclude that up to conjugacy and inverses, the only elements ofP_n^pR_P²qof order 4 areαand β, and that they only occur in the second case (i2 andr n1, ori 1 andrn). This completes the proof of the proposition.

As a corollary of the proof of Proposition 9, we are able to determine explicitly the orders of the torsion elements given by Proposition 7 as follows.

COROLLARY 11.Let n ^¥ 2. Then for all i 1, 2, and all r 0, 1, . . . ,n, the element y_i A_i^pn,r, 2r^{l,p^{l^qξ^2r{lω^p{l of B_n^pR_P²qis of order2l 2 gcd^pp, 2r^q.

Proof. If 1 ^¤ r ^¤ n2, we deduce from the above proof that y_i is of order 2l. Ifr 0 then l p, and y_i ω which is of order 2l. If i 1 and r n1 then l 1, and y_i ∆_n is indeed of order 2. Finally, ifi 2 andrn1 ori1 andrnthen p0 and l 2r. Hencey₁ ξ a which is of order 4n, and y₂ ξ b which is of order 4^pn1^q. The result follows.

Setl₁^pn,r^qgcd^p2r,nr^qandl₂^pn,r^qgcd^p2r,nr1^q. For 0^¤r^¤n1, clearly l₂^pn,r^ql₁^pn1,r^q, so it suffices to know the values ofl₁^pn,r^q. An alternative manner to express the order 2l₁^pn,r^qofy₁is given by the following:

COROLLARY 12.Let n^¥2. Let k

$

&

%

nr

2 if n and r are even nr otherwise.

Then

2l₁^pn,r^q

$

&

%

2 gcd^pn,r^q if k is odd 4 gcd^pn,k^q if k is even.

Proof. Suppose first that n and r are even, so k ⁿ₂^r. If k is odd then l₁^pn,r^q gcd^p2r,nr^q 2 gcd^pr,k^q 2 gcd

r

2,ⁿ₂^r gcd^pr,nr^q gcd^pn,r^q. If k is even then l₁^pn,r^q gcd^p2r,nr^q gcd^p2n,nr^q 2 gcd

n, ⁿ₂^r 2 gcd^pn,k^q, which yields the result in this case.

Now suppose that at least one of n and r is odd, so k nr. If k is odd then l₁^pn,r^q gcd^p2r,nr^q gcd^pr,nr^q gcd^pn,r^q. Ifk is even then both n and r are odd, andl₁^pn,r^qgcd^p2r,nr^q2 gcd^pr,nr^q2 gcd^pn,nr^q2 gcd^pn,k^q.

4 Virtually cyclic groups

We start this section by recalling the definition of a virtually cyclic group.

DEFINITION13. A group isvirtually cyclicif it contains a cyclic subgroup of finite index.

REMARKS 14.

(a) Every finite group is virtually cyclic.

(b) Every infinite virtually cyclic group contains a normal subgroup of finite index.

The following criterion is due to Wall:

THEOREM 15 ([P, W]).Let G be a group. Then the following are equivalent.

(a) G is a group with two ends.

(b) G is an infinite virtually cyclic group.

(c) G has a finite normal subgroup F such that G^{F isZ_orZ₂Z_2. Equivalently, G is of the form:

(i) FZ_{, or}

(ii) G_1F G₂, where^rG_i : F^s 2for i 1, 2.

If a virtually cyclic group G satisifies (i), we shall say that it is ofType I, while if it satisifies (ii), we shall say that it is ofType II.

We have already obtained the finite subgroups of P_n^pR_P²qin Proposition 1. Theo- rem 15 allows us to establish a list of their possible infinite virtually cyclic subgroups.

However, the difficulty is to decide whether the groups belonging to this list are effec- tively realised as subgroups ofP_n^pR_P²q.

The following lemma gives a practical criterion for deciding whether a given infinite group is virtually cyclic of Type II, and shall be applied frequently in what follows.

LEMMA16.Let GG_1FG₂be a virtually cyclic group of Type II, and letϕ: G_1FG₂^ÝÑH be a homomorphism such that the restriction of ϕto each G_iis injective. Then ϕis injective if and only ifϕ^pG^qis infinite.

Proof. Clearly the given condition is necessary. So suppose that ϕ^pG^q is infinite. By Theorem 15, we have the short exact sequence

1^ÝÑF^ÝÑG_1FG₂ ^ÝÑZ₂Z₂ÝÑ1.

Letx(resp.y) denote the generator of the first (resp. second) copy ofZ_{2. Then}

A

xy¹

E

Zis a normal subgroup ofZ₂Z_{2, and}Z₂Z₂ZZ_{2, where}Z₂may be taken as being generated by the

A

xy¹

E

-coset ofx.

Let Kbe the preimage of

A

xy¹

E

under the projectionG ^ÝÑZ₂Z₂. Then we have a short exact sequence

1^ÝÑF^ÝÑK ^ÝÑZ ÝÑ1,

and thusK FZis virtually cyclic of Type I, and is of index 2 inG. This gives rise to the following commutative diagram of short exact sequences:

1

1

1 ^//F ^{ //}K_{ _} ^//

Z ^//

 _

1

1 ^//F ^{ //}G ^//

Z₂Z₂ ^//

1 Z₂

Z₂

1 1

Let w ^P G₁^zF, and consider the image of w inG_1FG₂, which by abuse of notation we also denote byw. Since wis of finite order it cannot belong toK (for then it would have to be mapped onto 0 in Z, which is impossible), and hence G K^²wK. So by hypothesis,ϕ^pK^qis infinite.

Notice that the result that we are aiming to prove is true forKi.e. if we considerϕ|_K, and suppose that the restriction of ϕtoF is injective then the fact that ϕ^pK^q is infinite implies that ϕ|_K is injective. Indeed, identifyingKwith FZ_{, letk}px,m^qP Kbelong to the kernel of ϕ|_K, where x ^P F and m ^P Z_{. Since ϕ}pK^q is infinite, ϕ|Z is injective. If m 0 thene_H ϕ^pk^q ϕ^ppx, 0^qq.ϕ^ppe_G,m^qq. But ϕ^ppe_G,m^qqis of infinite order, while ϕ^ppx, 0^qqis of finite order, a contradiction. Som0, xe_F by injectivity of ϕ|_F, and so ϕ|_K is injective.

NowF ^€G₁, soϕ|_Fis injective. It follows then from the previous paragraph thatϕ|_K is injective. Furthermore, ϕ|_wK is injective since any two elements differ by an element ofK.

Finally, to prove the result, it suffices to show that for allk,k^{1 P} K, ϕ^pk^q ϕ^pwk^1q, or equivalently that for allk^{2 P}K, ϕ^pw^qϕ^pk^2q. Suppose on the contrary that there exists k^{2 P} Ksuch thatϕ^pw^q ϕ^pk^2q. Then ϕ^pw^2q ϕ^pk^22q. Noww^{2 P} KsinceKis of index 2 inG, and sow²k²²by injectivity of ϕ|_K. Thusk^{2 P}Kis of finite order, and so belongs toF. Hencew,k^{2 P} G₁,wk² (asw^P G₁^zF), and so ϕ^pw^q ϕ^pk^2qby injectivity ofϕ|_G1. This yields a contradiction. Hence ϕis injective.

5 Virtually cyclic subgroups of P

R _P

We now turn to the study of the virtually cyclic subgroups ofP_n^pR_P²q. AsP₁^pR_P²qZ₂ and P₂^pR_P²q Q₈, it is trivial to determine their virtually cyclic subgroups. We thus suppose from now on that n ^¥ 3. Since the structure of the finite subgroups differ for n 3 and n ^¥ 4, we treat these two cases separately. Further, by Proposition 1, up to isomorphism, we already know their finite subgroups. So in what follows, we shall seek their infinite virtually cyclic subgroups.