Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane

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Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere

and the projective plane

Daciberg Gonçalves, John Guaschi

To cite this version:

Daciberg Gonçalves, John Guaschi. Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane. Mathematische Zeitschrift, Springer, 2013, 274 (1-2), pp.667-683. �10.1007/s00209-012-1090-0�. �hal-00664589�

Minimal generating and normally generating sets for the braid and mapping class groups of D ² _, S ²

and R _P ²

DACIBERG LIMA GONÇALVES Departamento de Matemática - IME-USP, Caixa Postal 66281 - Ag. Cidade de São Paulo,

CEP: 05314-970 - São Paulo - SP - Brazil.

e-mail: [email protected] JOHN GUASCHI

Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen Basse-Normandie BP 5186,

14032 Caen Cedex, France.

e-mail: [email protected] 6th January 2012

Abstract

We consider the (pure) braid groups B_n^pM^qand P_n^pM^q, where M is the2-sphereS²_{or the} real projective planeR_P². We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves on results of Berrick and Matthey in the case ofS²_, and extends them in the case ofR_P². We begin by recalling the situation for the Artin braid groups (M D²). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for MS²_orR_P², the induced action of Bn^pM^qon H₃^pF^ƒn^pM^q;Zqis trivial, Fn^pM^qbeing the n^thconfiguration space of M.

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 p}p1, . . . ,p_n^q

p_i ^P Mand p_i p_j ifi j⁽.

2010 AMS Subject Classification: 20F36 (primary)

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

The symmetric groupSnonnletters acts freely onFn^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 Dn^pM^q). We thus obtain a natural short exact sequence

1^ÝÑP_n^pM^qÝÑB_n^pM^qÝπÑS_n ^ÝÑ1. (1) If D² „ R_P² is a topological disc, there is a group homomorphism ι: B_n ^ÝÑB_n^pM^q induced by the inclusion. Ifβ^P B_n then we shall denote its imageι^pβ^qsimply byβ.

Together with the 2-sphere S², the braid groups of the real projective plane R_P² are of particular interest, notably because they have non-trivial centre [VB, GG2], and torsion elements [VB, Mu]. Indeed, Fadell and Van Buskirk showed thatS²_andR_P²_are the only compact, connected surfaces whose braid groups have torsion. Let us recall briefly some of the properties of the braid groups of these two surfaces as well as those of the 2-discD²[A1, A2, FVB, GVB, GG2, H, MK, VB].

Let n ^P N. The Artin braid groupB_n, which may be interpreted as the braid group ofD², is generated byσ1, . . . ,σ_n1that are subject to the following relations:

σ_iσ_jσ_jσ_iif|ij| ^¥2 and 1^¤i,j^¤n1 (2) σ_iσ_{i 1}σ_i σ_{i 1}σ_iσ_{i 1}for all 1^¤i^¤n2. (3) The sphere braid group B_n^pS²q is also generated by σ1, . . . ,σ_n1, subject to the rela- tions (2) and (3), as well as the following ‘surface relation’:

σ1σ_n2σ_n²1σ_n2σ1 1. (4) Consequently, B_n^pS²qis a quotient ofB_n. The first three sphere braid groups are finite:

B1^pS²q is trivial, B2^pS²q is cyclic of order 2, and B3^pS²q is isomorphic to the dicyclic group of order 12 (the semi-direct productZ₃Z₄ with non-trivial action). Forn ^¥4, B_n^pS²qis infinite. The Abelianisation ofB_n (resp. B_n^pS²q) is isomorphic toZ _{(resp. the} cyclic groupZ₂

pn1^q), where the Abelianisation homomorphism identifies all of the σ_i to a single generator ofZ_{(resp. of}Z₂

pn1^q).

It is well known that B_n is torsion free. Gillette and Van Buskirk showed that if n^¥3 andk ^PN_thenBnpS²qhas an element of orderkif and only ifkdivides one of 2n, 2^pn1^qor 2^pn2^q[GVB]. The torsion elements ofB_n^pS²qwere later characterised by Murasugi:

THEOREM 1 (Murasugi [Mu]).Let n ^¥ 3. Then up to conjugacy, the torsion elements of B_n^pS²qare precisely the powers of the following three elements:

(a) α0σ1σn2σn1(which is of order2n).

(b) α1 σ1σ_n2σ_n²

1(of order2^pn1^q).

(c) α2 σ₁σ_n3σ_n²

2(of order2^pn2^q).

The three elementsα0, α1andα2are respectivelyn^th,^pn1^qthand^pn2^qthroots of

∆²_n, where∆²_n is the so-called ‘full twist’ braid ofB_n^pS²q, defined by∆²_n pσ1σ_n1^qn. In other words:

αⁿ₀ αⁿ₁¹α₂ⁿ² ∆²_n. (5)

SoB_n^pS²qadmits finite cyclic subgroups isomorphic toZ_2n,Z₂

pn1^qandZ₂

pn2^q. In [GG3, Theorem 3], we showed thatB_n^pS²qis generated byα0andα1. Ifn^¥3,∆²_n is the unique element ofB_n^pS²qof order 2, and it generates the centre of B_n^pS²q. It is also the square of theGarside element(or ‘half twist’) defined by:

∆_n pσ1σ_n1^qpσ1σ_n2^qpσ1σ2^qσ1. (6) Now let us turn to the braid groups ofR_P². A presentation is given by:

PROPOSITION 2 ([VB]).The following constitutes a presentation of the group B_n^pR_P²q: generators: σ₁, . . . ,σ_n1,ρ₁, . . . ,ρ_n.

relations:

(i) relations (2) and (3).

(ii) σ_iρ_j ρ_jσ_ifor j i,i 1.

(iii) ρ_{i 1} σ_i¹ρ_iσ_i¹for1^¤i^¤n1.

(iv) ρ_i ¹₁ρ_i ¹ρ_i 1ρ_i σ_i²for1^¤i ^¤n1.

(v) ρ²₁ σ₁σ₂σ_n2σ_n²

1σ_n2. . .σ₂σ₁.

For n ^¥ 2, the Abelianisation of B_n^pR_P²q is isomorphic to Z₂`Z_{2, the σi (resp.}

ρ_j) being identified to a generator of the first (resp. second factor). A presentation of P_n^pR_P²q was given in [GG4, Theorem 4] (note however that the generators ρ_i given there are slightly different from those of Van Buskirk). The first two braid groups of R_P² are finite: B1^pR_P^2q _P1^pR_P^2q Z_{2, P2}^pR_P^2q is isomorphic to the quaternion group Q₈ of order 8, and B₂^pR_P²q is isomorphic to the generalised quaternion group Q16 of order 16. Forn ^¥ 3, B_n^pR_P²qis infinite. The finite order elements of B_n^pR_P²q

were also characterised by Murasugi [Mu, Theorem B], however their orders were not clear, even for elements of P_n^pR_P²q. In [GG2, Corollary 19 and Theorem 4], we proved that for n ^¥ 2, the torsion of P_n^pR_P²qis 2 and 4, and that of B_n^pR_P²qis equal to the divisors of 4nand 4^pn1^q, and in [GG5, Section 3] we simplified somewhat Murasugi’s characterisation.

Following [GG2, Sections 3 and 4], set

#

a ρnσ_n1σ1

b ρ_n1σ_n2σ1 (7)

and ^#

Aaⁿ ρ_nρ1

Bbⁿ¹ ρ_n1ρ₁. (8)

These elements play an important rôle in the analysis of B_n^pR_P²q, and in particular in the study of many of its finite subgroups. By [GG2, Proposition 26],aandbare of order 4n and 4^pn1^qrespectively, so Aand B are of order 4. Further, by [GG5, Proposition 10], any element of order 4 in P_n^pR_P²qis conjugate via an element of B_n^pR_P²qto A or B. However, the number of conjugacy classes inP_n^pR_P²qof order 4 elements was not known. A naïve upper bound for this number, 2n!, may be obtained by multiplying the number of conjugacy classes inP_n^pR_P²qby^rB_n^pR_P²q : P_n^pR_P²qs. In Proposition 11, we shall compute the exact value.

IfΓis a group, letΓ^Ab denote its Abelianisation. ForX a subset ofΓ, let^xX^ydenote the subgroup of Γ generated by X, and let^xxX^yy denote the normal closure of X in Γ. Then Γ is generated(resp. normally generated) by X if Γ xX^y (resp. Γ xxX^yy). It is a natural question as to whetherΓxX^yis (normally) generated by a finite subset or not.

If it is, one can ask the following questions:

Question 1: compute

G^pΓqmin^t|X| |^Γis generated byX^u, and NG^pΓqmin^t|X| |^Γis normally generated byX^u, the minimal number of elements among all (normal) generating sets ofΓ.

Question 2: we can refine Question 1 if we impose additional constraints on the ele- ments ofX. We shall say that a groupΓistorsion generated(resp.normally torsion gener- ated) if there exists a subset X of elements ofΓof finite order such that Γ xX^y(resp.

Γ xxX^yy). If there exists a finite set X satisfying this property then one may ask to compute:

TG^pΓqmin^t|X| |^Γis torsion generated byX^u, and NTG^pΓqmin^t|X| |^Γis normally torsion generated byX^u,

the minimal number of elements among all (normal) generating sets ofGconsisting of finite order elements.

We have the following implications between the various notions:

torsion generated^ùñgenerated^ùñnormally generated

torsion generated^ùñnormally torsion generated^ùñnormally generated.

These relations imply that if the given numbers are defined for a groupΓthen:

NG^pΓq¤G^pΓq¤TG^pΓq and (9)

NG^pΓq¤NTG^pΓq¤TG^pΓq. (10) As a special case, recall from [BMa, BMi] that ifk ^¥ 2, Γis said to be strongly k-torsion generatedif there exists an elementg_k ^P Γof orderksuch thatΓxxg_k^yy. In other words, Γ is strongly k-torsion generated for some k ^P N if and only if NTG^pΓq 1. In [BMa], Berrick and Matthey considered the problem of strong torsion generation for various groups, among them the braid groups of S² _and R_P², and they proved the following result.

PROPOSITION 3 ([BMa, Proposition 5.3]).The normal closure of the element α1 has index gcd^pn, 2^qin B_n^pS²q. In particular, for n odd, B_n^pS²qis strongly2^pn1^q-torsion generated.

The question of the strong k-torsion generation of B_n^pR_P²qis mentioned in [BMa, page 923], although no result along the lines of Proposition 3 is given. The aim of this paper is to determine in general the numbers G^pΓqand NG^pΓq where Γ B_n^pM^q for M D²_,S² _orR_P², and the numbers TG^pΓqand NTG^pΓqwhere Γis one of B_n^pS²q or B_n^pR_P²q, as well as to find generating sets that realise these quantities. In a similar spirit, we shall also discuss these problems for the corresponding pure braid groups and their Abelianisation. ForB_n, the results are well known, but since we were not able to find a proof in the literature, we shall discuss this case briefly in Section 2.

PROPOSITION 4.Let n^¥2.

(a) G^pB_n^q

#

1 if n2

2 if n^¥3. Furthermore, for all n^¥2,^xxσ1^yy B_n, soNG^pB_n^q1.

(b) NG^pP_n^qG^pP_n^qG^pP_n^Abqn^pn1^q{2.

We turn to the case of the sphere in Section 3. The following result extends that of Proposition 3, and also treats the case ofP_n^pS²q. Note that ifn^¥3,∆²_n is the unique tor- sion element ofP_n^pS²q, and so ifn^¥4,P_n^pS²qcannot be (normally) torsion generated.

THEOREM 5.Let n^¥3, and for i0, 1, 2, letα_ibe as defined in Theorem 1.

(a) G^pB_n^pS²qq2,NG^pB_n^pS²qq1andTG^pB_n^pS²qq2.

(b) If n is even, there is no torsion element of B_n^pS²qwhose normal closure is B_n^pS²q. Further- more, B_n^pS²q{xxα₁^yyZ_2.

(c) NTG^pB_n^pS²qq

#

1 if n is odd 2 if n is even.

(d) The quotient of Bn^pS²qby the normal closure of eitherα0orα2is isomorphic toZ_n

1, unless n3and i2, in which case B3^pS²q{xxα2^yyS3.

(e) G^pP_n^pS²qqNG^pP_n^pS²qqG^pP_n^pS²q

Ab

qpn^pn3^q 2^q{2.

Part (c) includes Berrick and Matthey’s result given in Proposition 3.

In Section 4, we consider the braid groups ofR_P², and obtain the following results:

THEOREM 6.Let n^¥2, and let a and b be as given in equation (7).

(a) The group Bn^pR_P²qis generated by^ta,b^u, andG^pBn^pR_P²qqTG^pBn^pR_P²qq2.

(b) The normal closure of any element of B_n^pR_P²q is a proper subgroup of B_n^pR_P²q, and NG^pB_n^pR_P²qq NTG^pB_n^pR_P²qq 2. In particular, B_n^pR_P²q is not strongly k-torsion generated for any k^P N_.

(c) The quotient of B_n^pR_P²qby either^xxa^yyor by^xxb^yyis isomorphic toZ_2.

(d) The group P_n^pR_P²qis torsion generated by the following set of torsion elements:

Y

!

aⁿ,bⁿ¹,a¹bⁿ¹a, . . . ,a^pn2qbⁿ¹aⁿ²

)

(11) of order4. Further,

G^pP_n^pR_P²qqNG^pP_n^pR_P²qqTG^pP_n^pR_P²qqNTG^pP_n^pR_P²qqn. (12) In particular, P_n^pR_P²q cannot be normally generated by any subset consisting of less than n elements.

Part (b) thus answers negatively the underlying question of [BMa] concerning the strongk-torsion generation ofB_n^pR_P²q.

In Section 5, Proposition 4 and Theorems 5 and 6 will be applied to obtain similar results for the mapping class and pure mapping class groups of the given surfaces. As another application, in Section 6 we prove the triviality of the action on homology of the universal covering of the configuration space of the braid groups ofS²_andR_P²_: PROPOSITION 7.Let M be equal either toS²_{or to}R_P²_{, let n}¥3if M S²_{and n} ¥2if M R_P², and let H be any subgroup of B_n^pM^q. Then the induced action of H on H₃^pF^ƒ_n^pM^q;ZqZ is trivial. In particular the action of P_n^pM^qand B_n^pM^qon H3^pF^ƒ_n^pM^q;Zqis trivial.

Acknowledgements

This work took place during the visits of the first author to the Laboratoire de Math- ématiques Nicolas Oresme, Université de Caen Basse-Normandie during the period 5^th November – 5^th December 2010, and of the second author to the Departmento de Matemática do IME-Universidade de São Paulo during the periods 12^th – 25^th April and 4^th – 26^th October 2010, and 24^th February – 6^th March 2011. It was supported by the international Cooperation CNRS/FAPESP project number 09/54745-1, by the ANR project TheoGar project number ANR-08-BLAN-0269-02, and by the FAPESP “Projecto Temático Topologia Algébrica, Geométrica e Differencial” number 2008/57607-6.

2 Generating sets of B

and P

It is a classical fact that Bn and Pn are finitely presented, a presentation of the latter being given in [H, Appendix 1]. Although the results of Proposition 4 concerning the Artin braid groups are well known, we were not able to find an explicit statement in the literature, so we give a brief account here. Before doing so, we state the following result which allows us to compare the minimal number of generators of a group and its Abelianisation, and which will prove to be useful in what follows.

PROPOSITION8.LetΓ₁,Γ₂be groups, and letϕ: Γ₁ÝÑΓ₂be a surjective group homomorph- ism. IfΓ₁ is finitely generated then so isΓ₂, and we have the inequalitiesG^pΓ₁q ¥G^pΓ₂qand NG^pΓ₁q ¥NG^pΓ₂q. In particular, if Γ₂ Γ₁^Ab and ϕis Abelianisation,G^pΓ₁q ¥NG^pΓ₁q ¥

G^pΓ₁^Abq.

Proof. Straightforward.

We now prove Proposition 4.

Proof of Proposition 4.

(a) If n 2 then B2 Z, and the statement follows easily. The case n ^¥ 3 is a con- sequence of [MK, Chapter 2, Section 2, Exercise 2.4] of which a generalisation may be found in [GG7, Lemma 29]. Indeed, by induction we have:

pσ1σ2σ_n1^qiσ1^pσ1σ2σ_n1^qi

σ_i 1 fori 1, . . . ,n2,

which implies thatB_n ^xσ₁,σ₁σ2σ_n1^y. ButB_n is non cyclic, and so G^pB_n^q2. This calculation also shows thatBn ^xxσ1^yy, and thus NG^pBn^q1.

(b) Using the presentation of P_n given in [H, Appendix 1], we see that that P_n^Ab is a free Abelian group of rankn^pn1^q{2 with a basis comprised of theΓ₂pP_n^q-cosets of the standard generators

A_i,j σ_j1σ_{i 1}σ_i²σ_i¹₁σ_j1, 1 ^¤i  j ^¤n, (13) ofP_n, and denoted in [H] bya_i,j (hereΓ₂pP_n^qdenotes the commutator subgroup ofP_n).

Thus G^pP_n^q¥NG^pP_n^q¥G^pP_n^Abqn^pn1^q{2 by Proposition 8. ButP_nis generated by theA_i,j, son^pn1^q{2^¥G^pP_n^q, and the result follows.

3 Generating sets of B

S

and P

S

Since the groupB_n^pS²qis a quotient ofB_nandB_n^pS²qis not cyclic ifn^¥3, the statement of Proposition 4(a) remains true if we replaceBn byBn^pS²q. But for the case of Bn^pS²q, we can refine our analysis by looking at generating sets consisting of elements of finite order. It was proved in [GG3, Theorem 3] that B_n^pS²q xα0,α₁^y, α0 and α₁ being the elements of Theorem 1. By Proposition 3, if nis odd, Bn^pS²q xxα1^yy, and thus Bn^pS²q

is strongly 2^pn1^q-torsion generated in this case. If n is even, the same proposition shows that^xxα₁^yy is a subgroup of B_n^pS²qof index 2. In Theorem 5, we improve these results somewhat. We now proceed with the proof of that theorem.

Proof of Theorem 5.

(a) Propositions 4 and 8 imply that G^pB_n^pS²qq 2 and NG^pB_n^pS²qq 1, using the fact that Bn^pS²q is non cyclic and is a quotient of Bn. Since Bn^pS²q xα0,α1^y [GG3, Theorem 3], it then follows that TG^pB_n^pS²qq2.

(b) Equations (2), (3) and (4) imply that for n ^¥ 2, ^pB_n^pS²qq

Ab

Z₂

pn1^q, where the Abelianisation homomorphismϕ: Bn^pS²qÝÑpBn^pS²qq

Abidentifies the standard gener- atorsσ1, . . . ,σ_n1ofBn^pS²qto the generator 1 ofZ₂

pn1^q, and sendsα0,α1andα2ton1, nandn1 respectively. Ifi ^Pt0, 2^u, or ifi1 andnis even then^xϕ^pα_i^qypB_n^pS²qq

Ab, and so ^xxα_i^yy B_n^pS²q. The first part of the statement is then a consequence of The- orem 1, and the second part follows from Proposition 3.

(c) If n is odd, the result follows from Proposition 3, while if n is even, we see by equation (9) that NTG^pB_n^pS²qq 2 because G^pB_n^pS²qq 2 and NTG^pB_n^pS²qq 1 by part (b).

(d) We first treat the exceptional case n 3 andi 2. In this case, α2 σ₁² is a non- trivial torsion element of P3^pS²q, so is equal to ∆²₃, which generates P3^pS²q. The fact that B3^pS²q{xxα2^yy S3is a consequence of equation (1). So suppose thati ^{P t}0, 2^u, but with i 0 if n 3. The map which sends each σ_j, j 1, . . . ,n1, to the generator 1 of Z_n

1 extends to a surjective homomorphism ψ: B_n^pS²qÝÑZ_n

1. Since ψ^pα_i^q 0 and the target is Abelian, we see that ^xxα_i^{yy €} Ker^pψ^q and thus ψ factors through B_n^pS²q{xxα_i^yy. In particular, the order of B_n^pS²q{xxα_i^yyis at least n1. Ifi 0, the fact that B_n^pS²q xα₀,α₁^y implies that B_n^pS²q{xxα₀^yy is generated by the coset of α₁, so is cyclic. Using equation (5), it follows that Bn^pS²q{xxα0^yy is of order at mostn1, from which we deduce that it is cyclic of order exactly n1. Now suppose thati 2, so n ^¥4, and let Q B_n^pS²q{xxα₂^yy. By abuse of notation, we shall denote the projection of an elementx ^PBn^pS²qinQbyx. We will show that

σ1 σ_n1inQ. (14)

To do so, we shall prove by induction thatσ_n2i σn2inQfor all 1^¤i ^¤n3. For the casei 1, we first multiply the relation α2 1 inQon the left-hand side byσ_n2, so:

σn2 σn2α2 σn2σ1σ2σn3σn2σn2σ1σ2σ_n4σn2σn3σn2σn2

σ1σ2σ_n3σ_n2σ_n3σ_n2α2σ_n¹

2σ_n3σ_n2 using equation (3),

which in Q implies thatσn2 σ_n¹2σn3σn2, hence σn3 σn2. Now suppose that σ_n2i σ_n2inQ for some 1 ^¤ i ^¤ n4. Since ^pn2^qpn3i^q 1 i ^¥ 2, by

equations (2) and (3) and the induction hypothesis, we have:

σn2 σ_n3iσn2σ_n¹3i σ_n3iσ_n2iσ_n¹3iσ_n¹2iσ_n3iσ_n2i

σ_n¹2σ_n3iσ_n2 σ_n3i,

from which it follows that σ₁ σ_n2 in Q by induction. Since n ^¥ 4, in Q we obtain similarly:

σ_n1 σ_n3σ_n1σ_n¹3σ_n2σ_n1σ_n¹2σ_n¹

1σ_n2σ_n1 σ_n¹

1σ_n3σ_n1σ_n3, which yields equation (14). Denoting the common elementσ_i inQbyσ, we see thatQ is cyclic, generated byσ, and thatσⁿ¹1 inQbecauseα2is trivial in Q. This implies thatQis a quotient of Z_n

1, and since Qsurjects ontoZ_n

1, it follows from above that Q Z_n

1as required.

(e) By [GG1, Theorem 4(i)], we have the following isomorphism:

P_n^pS²qP_n3^pS²ztx₁,x₂,x₃^uq`Z_{2, (15)} where theZ₂-factor is generated by∆²_n. Using the presentation given in [GG3, Propos- ition 7], P_n3^pS²ztx₁,x₂,x₃^{u q} admits a generating set X A_i,j

4^¤ j^¤n, 1^¤i  j⁽ consisting of ^pn3^qpn 2^q{2 elements, where A_i,j is as defined in equation (13). For eachj4, . . . ,n, the surface relation

±j1 i1A_i,j

±n

kj 1A_j,k 1 allows us to delete the generator A_1,j from X, yielding a generating set X¹ A_i,j

4^¤ j^¤n, 2^¤i  j⁽ consisting ofn^pn3^q{2 elements, so

G^pP_n^pS²qq¤pn^pn3^q 2^q{2 (16) by equation (15). Furthermore, the set X¹ is just now subject to the remaining Artin braid relations (the those given at the top of [GG3, page 385]) of the presentation of P_n3^pS²ztx₁,x₂,x₃^uq, rewritten in terms of the elements ofX¹. These relations may be written as commutators of elements of X¹, and so collapse under Abelianisation. Thus

pPn3^pS²ztx1,x2,x3^uqqAb

Z^npn3q{2, for which a basis is given by the cosets of the elements of X¹. We deduce from equation (15) that ^pP_n^pS²qq

Ab

Z^npn3q{2`Z_{2, and} so G^ppPn^pS²qq

Ab

q pn^pn3^q 2^q{2. The result then follows from Proposition 8 and equation (16).

4 Generating sets of B

R _P

and P

R _P

We now consider the case of the braid groups of the projective plane, and prove The- orem 6.

Proof of Theorem 6.

(a) As we mentioned just after equation (7),aandbare torsion elements. SinceB_n^pR_P²q

is non cyclic for n ^¥ 2 (its Abelianisation isZ₂`Z₂ by Proposition 2), we thus have G^pBn^pR_P²qq¥2. LetLbe the subgroup ofBn^pR_P²qgenerated by^ta,b^u. By equation (7), we have:

ab^{1 p}ρ_nσ_n1σ1^qpρ_n1σ_n2σ1^q1

ρ_nσ_n1ρ_n¹

1σ_n¹

1,