Table Of ContentMinimal generating and normally generating sets
D S
2 2
for the braid and mapping class groups of ,
R
and P2
2 DACIBERG LIMA GONÇALVES
1
0 Departamento de Matemática - IME-USP,
2
Caixa Postal 66281 - Ag. Cidade de São Paulo,
n
a CEP: 05314-970 - São Paulo - SP - Brazil.
J
e-mail: [email protected]
1
3
JOHN GUASCHI
]
T
Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139,
G
. Université de Caen Basse-Normandie BP 5186,
h
t 14032 Caen Cedex, France.
a
m
e-mail: [email protected]
[
1 6th January 2012
v
2
8
4
Abstract
6
1. Weconsiderthe(pure)braidgroups B pMqand P pMq,where M isthe2-sphere S2 orthe
n n
0 realprojectiveplaneRP2. Wedeterminetheminimalcardinalityof(normal)generatingsets
2
X ofthesegroups, firstwhenthere isnorestriction on X,andsecondly when X consists of
1
: elements of finiteorder. Thisimproves on results of Berrick and Matthey in the case of S2,
v
i andextendstheminthecaseofRP2. WebeginbyrecallingthesituationfortheArtinbraid
X groups (M “ D2). As applications of our results, we answer the corresponding questions
ar fortheassociatedmappingclassgroups,andweshowthatfor M “ S2 orRP2,theinduced
actionofB pMqonH pF pMq;Zqistrivial, F pMqbeingthenthconfigurationspaceof M.
n 3 n n
Č
1 Introduction
The braid groups B of the plane were introduced by E. Artin in 1925 [A1, A2]. Braid
n
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
N
beacompact,connectedsurface,andletn P . Wedenotethesetofallorderedn-tuples
ofdistinct points of M, known asthe n configurationspaceof M, by:
(cid:12)
FnpMq “ pp1,...,pnq (cid:12)pi P M and pi ‰ pj ifi ‰ j .
2010AMSSubjectClassifica tion: 20F36(primary) (
1
Configuration spaces play an important rôle in several branches of mathematics and
have beenextensively studied, see [CG, FH]forexample.
ThesymmetricgroupS onnlettersactsfreelyon F pMqbypermutingcoordinates.
n n
The corresponding quotient will be denoted by D pMq. The n pure braid group P pMq
n n
(respectively the n braid group B pMq) isdefined tobe the fundamental group of F pMq
n n
(respectively of D pMq). Wethus obtain anatural short exact sequence
n
π
1 ÝÑ P pMq ÝÑ B pMq ÝÑ S ÝÑ 1. (1)
n n n
If D2 Ď RP2 is a topological disc, there is a group homomorphism ι: B ÝÑ B pMq
n n
induced bythe inclusion. If β P B then weshall denote its image ιpβq simply by β.
n
Together with the 2-sphere S2, the braid groups of the real projective plane RP2
are of particular interest, notably because they have non-trivial centre [VB, GG2], and
torsionelements[VB,Mu]. Indeed,FadellandVanBuskirkshowedthatS2andRP2 are
the only compact, connected surfaces whose braid groups have torsion. Let us recall
brieflysome ofthe propertiesofthebraidgroups ofthesetwosurfacesaswellasthose
ofthe 2-discD2 [A1,A2, FVB, GVB, GG2, H, MK, VB].
N
Let n P . The Artin braid group B , which may be interpreted as the braid group
n
ofD2, isgenerated by σ ,...,σ thatare subject to the following relations:
1 n´1
σσ “ σσ if |i´j| ě 2and 1 ď i,j ď n´1 (2)
i j j i
σσ σ “ σ σσ for all1 ď i ď n´2. (3)
i i`1 i i`1 i i`1
The sphere braid group B pS2q is also generated by σ ,...,σ , subject to the rela-
n 1 n´1
tions (2)and (3), aswellasthe following ‘surface relation’:
σ ¨¨¨σ σ2 σ ¨¨¨σ “ 1. (4)
1 n´2 n´1 n´2 1
Consequently, B pS2q is a quotient of B . The first three sphere braid groups are finite:
n n
B pS2q is trivial, B pS2q is cyclic of order 2, and B pS2q is isomorphic to the dicyclic
1 2 3
Z Z
group of order 12 (the semi-direct product ¸ with non-trivial action). For n ě 4,
3 4
B pS2q is infinite. The Abelianisation of B (resp. B pS2q) is isomorphic to Z (resp. the
n n n
Z
cyclic group ), where the Abelianisation homomorphism identifies all of the σ
2pn´1q i
Z Z
to asingle generator of (resp. of ).
2pn´1q
It is well known that B is torsion free. Gillette and Van Buskirk showed that if
n
n ě 3and k P Nthen B pS2qhasanelementoforder k ifandonlyifk dividesoneof2n,
n
2pn´1q or 2pn´2q [GVB]. The torsion elements of B pS2q were later characterised by
n
Murasugi:
THEOREM 1 (Murasugi [Mu]). Let n ě 3. Then up to conjugacy, the torsion elements of
B pS2qare preciselythe powersof thefollowingthree elements:
n
(a) α “ σ ¨¨¨σ σ (whichisof order2n).
0 1 n´2 n´1
(b) α “ σ ¨¨¨σ σ2 (of order2pn´1q).
1 1 n´2 n´1
(c) α “ σ ¨¨¨σ σ2 (of order2pn´2q).
2 1 n´3 n´2
The three elements α , α and α are respectively n, pn´1q and pn´2q roots of ∆2,
0 1 2 n
where ∆2 is the so-called ‘full twist’ braid of B pS2q, defined by ∆2 “ pσ ¨¨¨σ qn. In
n n n 1 n´1
other words:
αn “ αn´1 “ αn´2 “ ∆2. (5)
0 1 2 n
2
SoB pS2qadmitsfinitecyclicsubgroupsisomorphictoZ ,Z andZ . In[GG3,
n 2n 2pn´1q 2pn´2q
Theorem3],weshowedthat B pS2qisgeneratedbyα andα . Ifn ě 3,∆2 istheunique
n 0 1 n
element of B pS2q of order 2, and it generates the centre of B pS2q. It is also the square
n n
ofthe Garsideelement(or ‘halftwist’) definedby:
∆ “ pσ ¨¨¨σ qpσ ¨¨¨σ q¨¨¨pσ σ qσ . (6)
n 1 n´1 1 n´2 1 2 1
Now letusturn tothe braid groups ofRP2. Apresentation isgiven by:
PROPOSITION 2([VB]). Thefollowingconstitutes apresentationof thegroup BnpRP2q:
generators: σ ,...,σ ,ρ ,...,ρ .
1 n´1 1 n
relations:
(i) relations (2)and (3).
(ii) σρ “ ρ σ for j ‰ i,i`1.
i j j i
(iii) ρ “ σ´1ρ σ´1 for 1 ď i ď n´1.
i`1 i i i
(iv) ρ´1 ρ´1ρ ρ “ σ2 for 1 ď i ď n´1.
i`1 i i`1 i i
(v) ρ2 “ σ σ ¨¨¨σ σ2 σ ...σ σ .
1 1 2 n´2 n´1 n´2 2 1
For n ě 2, the Abelianisation of B pRP2q is isomorphic to Z ‘Z , the σ (resp.
n 2 2 i
ρ ) being identified to a generator of the first (resp. second factor). A presentation of
j
P pRP2q was given in [GG4, Theorem 4] (note however that the generators ρ given
n i
there are slightly different from those of Van Buskirk). The first two braid groups of
RP2 are finite: B pRP2q “ P pRP2q – Z , P pRP2q is isomorphic to the quaternion
1 1 2 2
group Q of order 8, and B pRP2q is isomorphic to the generalised quaternion group
8 2
Q of order 16. For n ě 3, B pRP2q is infinite. The finite order elements of B pRP2q
16 n n
were also characterised by Murasugi [Mu, Theorem B], however their orders were not
clear, even for elements of P pRP2q. In [GG2, Corollary 19 and Theorem 4], we proved
n
that for n ě 2, the torsion of P pRP2q is 2 and 4, and that of B pRP2q is equal to the
n n
divisorsof4nand4pn´1q,andin[GG5,Section3]wesimplifiedsomewhatMurasugi’s
characterisation.
Following [GG2, Sections 3 and4], set
a “ ρ σ ¨¨¨σ
n n´1 1
(7)
b “ ρ σ ¨¨¨σ
# n´1 n´2 1
and
A “ an “ ρ ¨¨¨ρ
n 1
(8)
# B “ bn´1 “ ρn´1¨¨¨ρ1.
These elements play an important rôle in the analysis of B pRP2q, and in particular in
n
thestudyofmanyofitsfinitesubgroups. By[GG2,Proposition 26], aandbareoforder
4n and 4pn´1q respectively, so A and B are of order 4. Further, by [GG5, Proposition
10], any element of order 4 in P pRP2q is conjugate via an element of B pRP2q to A or
n n
B. However, the number of conjugacy classes in P pRP2q of order 4 elements was not
n
known. Anaïveupperboundforthisnumber,2n!,maybeobtainedbymultiplyingthe
number of conjugacy classes in P pRP2q by rB pRP2q : P pRP2qs. In Proposition 11, we
n n n
shall compute the exactvalue.
3
If Γ is a group, let ΓAb denote its Abelianisation. For X a subset of Γ, let xXy denote
the subgroup of Γ generated by X, and let xxXyy denote the normal closure of X in Γ.
Then Γ is generated (resp. normally generated) by X if Γ “ xXy (resp. Γ “ xxXyy). It is a
naturalquestionastowhetherΓ “ xXyis(normally)generatedbyafinitesubsetornot.
Ifitis, one can askthe following questions:
Question 1: compute
GpΓq “ mint|X| |Γ isgenerated by Xu, and
NGpΓq “ mint|X| |Γ isnormally generated by Xu,
Γ
the minimal numberofelementsamongall (normal) generating setsof .
Question 2: we can refine Question 1 if we impose additional constraints on the ele-
Γ
ments of X. We shall say that a group is torsiongenerated (resp. normally torsion gener-
ated) if there exists a subset X of elements of Γ of finite order such that Γ “ xXy (resp.
Γ “ xxXyy). If there exists a finite set X satisfying this property then one may ask to
compute:
TGpΓq “ mint|X| |Γ istorsion generated by Xu, and
NTGpΓq “ mint|X| |Γ isnormally torsion generated by Xu,
the minimal number of elements among all (normal) generating sets of G consisting of
finite order elements.
Wehave the following implications between the various notions:
torsion generated ùñgenerated ùñnormally generated
torsion generated ùñnormally torsion generated ùñnormally generated.
Γ
These relations implythatif the given numbersare defined fora group then:
NGpΓq ď GpΓq ď TGpΓq and (9)
NGpΓq ď NTGpΓq ď TGpΓq. (10)
As a special case, recall from [BMa, BMi] that if k ě 2, Γ is said to be strongly k-torsion
generatedifthereexistsanelementg P ΓoforderksuchthatΓ “ xxg yy. Inotherwords,
k k
Γ is strongly k-torsion generated for some k P N if and only if NTGpΓq “ 1. In [BMa],
Berrick and Matthey considered the problem of strong torsion generation for various
groups, among them the braid groups of S2 and RP2, and they proved the following
result.
PROPOSITION 3 ([BMa, Proposition 5.3]). The normal closure of the element α1 has index
gcdpn,2q in B pS2q. Inparticular,for n odd, B pS2qis strongly2pn´1q-torsiongenerated.
n n
The question of the strong k-torsion generation of B pRP2q is mentioned in [BMa,
n
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 GpΓq and NGpΓq where Γ “ B pMq for
n
M “ D2,S2 or RP2, and the numbers TGpΓq and NTGpΓq where Γ is one of B pS2q or
n
B pRP2q, as well as to find generating sets that realise these quantities. In a similar
n
spirit, we shall also discuss these problems for the corresponding pure braid groups
andtheirAbelianisation. For B ,theresultsarewellknown,butsincewewerenotable
n
to find a proof in the literature, we shalldiscuss this case brieflyin Section 2.
4
PROPOSITION 4. Let n ě 2.
1 if n “ 2
(a) GpB q “ Furthermore,for all n ě 2, xxσ yy “ B , soNGpB q “ 1.
n 1 n n
#2 if n ě 3.
(b) NGpP q “ GpP q “ GpP Abq “ npn´1q{2.
n n n
We turn to the case of the sphere in Section 3. The following result extends that of
Proposition 3,andalsotreatsthecaseof P pS2q. Notethatifn ě 3,∆2 istheuniquetor-
n n
sion elementof P pS2q,and soif n ě 4, P pS2q cannot be (normally) torsion generated.
n n
THEOREM 5. Let n ě 3, and for i “ 0,1,2,letαi be as definedinTheorem1.
(a) GpB pS2qq “ 2, NGpB pS2qq “ 1 and TGpB pS2qq “ 2.
n n n
(b) If n is even,thereisno torsionelementof B pS2q whosenormal closureis B pS2q. Further-
n n
more, B pS2q{xxα yy – Z .
n 1 2
1 if n isodd
(c) NTGpB pS2qq “
n
#2 if n iseven.
(d) ThequotientofB pS2qbythenormalclosureofeitherα orα isisomorphictoZ ,unless
n 0 2 n´1
n “ 3 and i “ 2,inwhichcase B pS2q{xxα yy – S .
3 2 3
(e) GpP pS2qq “ NGpP pS2qq “ GpP pS2qAbq “ pnpn´3q`2q{2.
n n n
Part (c) includesBerrickand Matthey’s result given in Proposition 3.
In Section 4,we consider the braid groups of RP2, andobtain the following results:
THEOREM 6. Let n ě 2, and let a and b be as giveninequation (7).
(a) The group B pRP2q isgenerated byta,bu, and GpB pRP2qq “ TGpB pRP2qq “ 2.
n n n
(b) The normal closure of any element of B pRP2q is a proper subgroup of B pRP2q, and
n n
NGpB pRP2qq “ NTGpB pRP2qq “ 2. In particular, B pRP2q is not strongly k-torsion
n n n
N
generatedfor any k P .
(c) Thequotient of B pRP2q by eitherxxayy or by xxbyy isisomorphictoZ .
n 2
(d) Thegroup P pRP2q istorsiongenerated by thefollowing setof torsionelements:
n
Y “ an,bn´1,a´1bn´1a,...,a´pn´2qbn´1an´2 (11)
! )
of order4. Further,
GpP pRP2qq “ NGpP pRP2qq “ TGpP pRP2qq “ NTGpP pRP2qq “ n. (12)
n n n n
In particular, P pRP2q cannot be normally generated by any subset consisting of less than n
n
elements.
Part (b) thus answers negatively the underlying question of [BMa] concerning the
strong k-torsion generation of B pRP2q.
n
In Section 5, Proposition 4 and Theorems 5 and 6 will be applied to obtain similar
results for the mappingclass 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 ofthe configuration spaceof the braid groups of S2 andRP2:
PROPOSITION 7. Let M beequaleithertoS2 ortoRP2,letn ě 3if M “ S2 andn ě 2if M “
RP2,andletHbeanysubgroupofB pMq. Thentheinducedactionof Hon H pF pMq;Zq – Z
n 3 n
Z
istrivial. In particularthe actionof P pMq and B pMq on H pF pMq; q istrivial.
n n 3 n
Č
Č
5
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
5th November – 5th December 2010, and of the second author to the Departmento de
Matemática do IME-Universidade de São Paulo during the periods 12th – 25th April
and 4th – 26th October 2010, and 24th February – 6th March 2011. It was supported by
theinternational Cooperation CNRS/FAPESPprojectnumber09/54745-1,bytheANR
project TheoGar project numberANR-08-BLAN-0269-02,and bythe FAPESP“Projecto
TemáticoTopologia Algébrica, Geométrica e Differencial” number2008/57607-6.
2 Generating sets of B and P
n n
It is a classical fact that B and P are finitely presented, a presentation of the latter
n n
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
resultwhichallowsustocomparetheminimalnumberofgeneratorsofagroupandits
Abelianisation, andwhich will prove to be useful in whatfollows.
PROPOSITION 8. LetΓ1,Γ2 begroups,andlet ϕ: Γ1 ÝÑ Γ2 beasurjectivegrouphomomorph-
ism. If Γ is finitely generated then so is Γ , and we have the inequalities GpΓ q ě GpΓ q and
1 2 1 2
NGpΓ q ě NGpΓ q. In particular, if Γ “ Γ Ab and ϕ is Abelianisation, GpΓ q ě NGpΓ q ě
1 2 2 1 1 1
GpΓ Abq.
1
Proof. Straightforward.
Wenow prove Proposition 4.
Proof of Proposition4.
Z
(a) If n “ 2 then B – , and the statement follows easily. The case n ě 3 is a con-
2
sequence of [MK, Chapter 2, Section 2, Exercise 2.4] of which a generalisation may be
found in [GG7, Lemma29]. Indeed,byinduction we have:
pσ σ ¨¨¨σ qiσ pσ σ ¨¨¨σ q´i “ σ for i “ 1,...,n´2,
1 2 n´1 1 1 2 n´1 i`1
which impliesthat B “ xσ ,σ σ ¨¨¨σ y. But B is non cyclic, and so GpB q “ 2. This
n 1 1 2 n´1 n n
calculation alsoshows that B “ xxσ yy, and thusNGpB q “ 1.
n 1 n
(b) Using the presentation of P given in [H, Appendix 1], we see that that P Ab is a
n n
freeAbeliangroupofranknpn´1q{2withabasiscomprisedoftheΓ pP q-cosetsofthe
2 n
standard generators
A “ σ ¨¨¨σ σ2σ´1 ¨¨¨σ , 1 ď i ă j ď n, (13)
i,j j´1 i`1 i i`1 j´1
of P , and denoted in [H] by a (here Γ pP q denotes the commutator subgroup of P ).
n i,j 2 n n
ThusGpP q ě NGpP q ě GpP Abq “ npn´1q{2byProposition 8. But P isgeneratedby
n n n n
the A , so npn´1q{2 ě GpP q, andthe result follows.
i,j n
6
3 Generating sets of B pS2q and P pS2q
n n
Sincethegroup B pS2qisaquotientof B and B pS2qisnotcyclicifn ě 3,thestatement
n n n
of Proposition 4(a) remains true if we replace B by B pS2q. But for the case of B pS2q,
n n n
we can refine our analysis bylooking at generating sets consisting of elementsof finite
order. It was proved in [GG3, Theorem 3] that B pS2q “ xα ,α y, α and α being the
n 0 1 0 1
elements of Theorem 1. By Proposition 3, if n is odd, B pS2q “ xxα yy, and thus B pS2q
n 1 n
is strongly 2pn´1q-torsion generated in this case. If n is even, the same proposition
shows that xxα yy is a subgroup of B pS2q of index 2. In Theorem 5, we improve these
1 n
results somewhat. Wenowproceed with the proof of thattheorem.
Proof of Theorem5.
(a) Propositions 4 and 8 imply that GpB pS2qq “ 2 and NGpB pS2qq “ 1, using the
n n
fact that B pS2q is non cyclic and is a quotient of B . Since B pS2q “ xα ,α y [GG3,
n n n 0 1
Theorem 3], itthen follows thatTGpB pS2qq “ 2.
n
(b) Equations (2), (3) and (4) imply that for n ě 2, pB pS2qqAb – Z , where the
n 2pn´1q
Abelianisationhomomorphism ϕ: B pS2q ÝÑ pB pS2qqAbidentifiesthestandardgener-
n n
atorsσ ,...,σ ofB pS2qtothegenerator1ofZ ,andsendsα ,α andα ton´1,
1 n´1 n 2pn´1q 0 1 2
n and n´1 respectively. Ifi P t0,2u, or ifi “ 1 and n is eventhen xϕpα qy ‰ pB pS2qqAb,
i n
and so xxα yy ‰ B pS2q. The first part of the statement is then a consequence of The-
i n
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 NTGpB pS2qq “ 2 because GpB pS2qq “ 2 and NTGpB pS2qq ‰ 1 by
n n n
part(b).
(d) We first treat the exceptional case n “ 3 and i “ 2. In this case, α “ σ2 is a non-
2 1
trivial torsion element of P pS2q, so is equal to ∆2, which generates P pS2q. The fact
3 3 3
that B pS2q{xxα yy – S is a consequence of equation (1). So suppose that i P t0,2u, but
3 2 3
with i “ 0 if n “ 3. The map which sends each σ, j “ 1,...,n´1, to the generator
j
1 of Z extends to a surjective homomorphism ψ: B pS2q ÝÑ Z . Since ψpα q “
n´1 n n´1 i
0 and the target is Abelian, we see that xxα yy Ă Kerpψq and thus ψ factors through
i
B pS2q{xxα yy. In particular, the order of B pS2q{xxα yy is at least n´1. If i “ 0, the fact
n i n i
that B pS2q “ xα ,α y implies that B pS2q{xxα yy is generated by the coset of α , so is
n 0 1 n 0 1
cyclic. Using equation (5), it follows that B pS2q{xxα yy is of order at most n´1, from
n 0
which we deduce that it is cyclic of order exactly n´1. Now suppose that i “ 2, so
n ě 4, and let Q “ B pS2q{xxα yy. By abuse of notation, we shall denote the projection
n 2
ofan element x P B pS2q in Q by x. Wewill showthat
n
σ “ ¨¨¨ “ σ in Q. (14)
1 n´1
To do so, we shall prove by induction that σ “ σ in Q for all 1 ď i ď n´3. For
n´2´i n´2
the case i “ 1, we first multiply the relation α “ 1 in Q on the left-hand side by σ ,
2 n´2
so:
σ “ σ α “ σ σ σ ¨¨¨σ σ σ “ σ σ ¨¨¨σ σ σ σ σ
n´2 n´2 2 n´2 1 2 n´3 n´2 n´2 1 2 n´4 n´2 n´3 n´2 n´2
“ σ σ ¨¨¨σ σ σ σ “ α σ´1 σ σ using equation (3),
1 2 n´3 n´2 n´3 n´2 2 n´2 n´3 n´2
which in Q implies that σ “ σ´1 σ σ , hence σ “ σ . Now suppose that
n´2 n´2 n´3 n´2 n´3 n´2
σ “ σ in Q for some 1 ď i ď n´4. Since pn´2q´pn´3´iq “ 1`i ě 2, by
n´2´i n´2
7
equations(2)and (3)and the induction hypothesis, wehave:
σ “ σ σ σ´1 “ σ σ σ´1 “ σ´1 σ σ
n´2 n´3´i n´2 n´3´i n´3´i n´2´i n´3´i n´2´i n´3´i n´2´i
“ σ´1 σ σ “ σ ,
n´2 n´3´i n´2 n´3´i
from which it follows that σ “ ¨¨¨ “ σ in Q by induction. Since n ě 4, in Q we
1 n´2
obtain similarly:
σ “ σ σ σ´1 “ σ σ σ´1 “ σ´1 σ σ “ σ´1 σ σ “ σ ,
n´1 n´3 n´1 n´3 n´2 n´1 n´2 n´1 n´2 n´1 n´1 n´3 n´1 n´3
which yields equation (14). Denoting the common element σ in Q by σ, we see that Q
i
is cyclic, generated by σ, and that σn´1 “ 1 in Q because α istrivial in Q. This implies
2
Z Z
that Q is a quotient of , and since Q surjects onto , it follows from above that
n´1 n´1
Z
Q – asrequired.
n´1
(e) By[GG1, Theorem 4(i)], we have the following isomorphism:
P pS2q – P pS2ztx ,x ,x uq‘Z , (15)
n n´3 1 2 3 2
where the Z -factor is generated by ∆2. Using the presentation given in [GG3, Propos-
2 n
ition 7], Pn´3pS2ztx1,x2,x3uq admits a generating set X “ Ai,j(cid:12)(cid:12) 4 ď j ď n, 1 ď i ă j
consisting of pn´3qpn`2q{2 elements, where A is as defined in equation (13). For
i,j
(
j´1 n
each j “ 4,...,n, the surface relation A A “ 1 allowsus to delete
i“1 i,j k“j`1 j,k
(cid:12)
the generator A from X, yielding a´generatin¯g´set X1 “ A¯ (cid:12) 4 ď j ď n, 2 ď i ă j
1,j ś ś i,j
consisting of npn´3q{2elements, so
(
GpP pS2qq ď pnpn´3q`2q{2 (16)
n
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 pS2ztx ,x ,x uq, rewritten in terms of the elements of X1. These relations may be
n´3 1 2 3
written as commutators of elements of X1, and so collapse under Abelianisation. Thus
pP pS2ztx ,x ,x uqqAb – Znpn´3q{2, for which a basis is given by the cosets of the
n´3 1 2 3
elements of X1. We deduce from equation (15) that pP pS2qqAb – Znpn´3q{2 ‘Z , and
n 2
so GppP pS2qqAbq “ pnpn ´3q`2q{2. The result then follows from Proposition 8 and
n
equation (16).
4 Generating sets of B pRP2q and P pRP2q
n n
We now consider the case of the braid groups of the projective plane, and prove The-
orem 6.
Proof of Theorem6.
(a) Aswementionedjustafterequation(7),aandbaretorsionelements. SinceB pRP2q
n
Z Z
is non cyclic for n ě 2 (its Abelianisation is ‘ by Proposition 2), we thus have
2 2
GpB pRP2qq ě 2. Let Lbethesubgroupof B pRP2qgeneratedbyta,bu. Byequation(7),
n n
we have:
ab´1 “ pρ σ ¨¨¨σ qpρ σ ¨¨¨σ q´1 “ ρ σ ρ´1 “ σ´1 ,
n n´1 1 n´1 n´2 1 n n´1 n´1 n´1
8
using relation (iii) of Proposition 2, so σ P L. From [GG2, page 777], ajσ a´j “
n´1 n´1
σ for all j “ 1,...,n´2, and so σ P L for all i “ 1,...,n´1. Hence ρ P L by
n´j´1 i n
equation (7), and again by[GG2, page 777],itfollows that
ajρ a´j “ ρ for all j “ 1,...,n´1, (17)
n n´j
and so ρ P L for all i “ 1,...,n. Proposition 2 implies that L “ B pRP2q, and so
i n
TGpB pRP2qq “ 2. Equation (9)then forces TGpB pRP2qq “ GpB pRP2qq “ 2.
n n n
(b) SincepB pRP2qqAb – Z ‘Z ,wehaveGppB pRP2qqAbq “ 2. Itfollowsfrompart(a),
n 2 2 n
Proposition 8 and equation (10) that NGpB pRP2qq “ NGpB pRP2qq “ 2, in particular
n n
xxxyy Ř B pRP2q forall x P B pRP2q.
n n
(c) Let x P ta,bu. By part (b), xxxyy ‰ B pRP2q. Since B pRP2q “ xa,by by part (a),
n n
B pRP2q{xxxyy is cyclic, so Abelian, and thus the projection B pRP2q ÝÑ B pRP2q{xxxyy
n n n
factors through pB pRP2qqAb – Z ‘Z ,from whichitfollowsthat B pRP2q{xxxyy – Z .
n 2 2 n 2
(d) Recall from [GG2, Corollary 19] that the torsion elements of P pRP2q are of order
n
2 or 4, and the only element of order 2 is the full twist. By arguments similar to those
of[GG4, Theorem 4],one may seethat P pRP2q isgenerated by
n
(cid:12)
A , ρ (cid:12) 1 ď i ă j ď n, 1 ď k ď n , (18)
i,j k
@ D
where A is given by equation (13) and ρ is that of Proposition 2 (beware however
i,j k
that the elements ρ of [GG4] differ somewhat from those of Proposition 2), and that
i
pP pRP2qqAb – Zn,whereundertheAbelianisationhomomorphismξ: P pRP2q ÝÑ Zn,
n 2 n 2
ρ is sent to the element p0,...,0, 1 ,0,...,0q of Zn, and for all 1 ď i ă j ď n,
k 2
kthposition
A is sent to the trivial element. lPoomrooopnosition 8 thus implies that NGpP pRP2qq ě
i,j n
GppP pRP2qqAbq “ n. In order to obtain equation (12), it suffices by equations (9)
n
and (10) to show that TGpP pRP2qq ď n. We achieve this by proving that the set Y
n
described by equation (11) consists of torsion elements and that it generates P pRP2q.
n
Thefirst assertion follows immediatelyfrom the factthat a and b are offinite order (re-
call from equation (8) that an and bn´1 are both of order 4, hence all of the elements of
Y are oforder 4). It remainsto showthat P pRP2q “ xYy. Todoso, first observe that by
n
the relation
ρ´1ρ´1ρ ρ “ A for all 1 ď i ă j ď n
j i j i i,j
given in [GG2, Lemma 17], it follows using equation (18) that P pRP2q is generated by
n
tρ ,...,ρ u(notethattheelementsdenotedby B in[GG2]aretheelements A ofthis
1 n i,j i,j
paper). From equation (8), AB´1 “ ρ , and applying equation (17), we see that for
n
j “ 0,1,...,n´1,
a´jpAB´1qaj “ A.a´jB´1aj “ ρ . (19)
n´j
Thus A, a´jBaj, j “ 0,1,...,n´1 “ xρ ,...,ρ y “ P pRP2q. But A “ ρ ¨¨¨ρ by
1 n n n 1
equation (8), and so
@ D
n´2
a´pn´1qB´1an´1 “ A´1ρ “ A´1pρ´1¨¨¨ρ´1qA “ A´1 an´2´jBaj`2´n.A´1 .A
1 2 n ¨ ˛
źj“0´ ¯
˝ ‚
9
by equation (19). We can thus remove a´pn´1qB´1an´1 from the list of generators, and
so P pRP2q “ xYy, which proves the first part of the statement. Since the elements ofY
n
are oforder 4, we obtain the inequalityTGpP pRP2qq ď n asrequired.
n
We finish this section by computing the number of conjugacy classes in B pRP2q of
n
elements of order 4 lying in P pRP2q. Before doing so, we determine the centraliser of
n
the elements a and b in B pRP2q. If Γ is a group and y P Γ, we denote the centraliser of
n
y in Γ by ZΓpyq, and the centre of Γ by ZpΓq. A sketch of the following result appeared
in[GG5,Remark24]. Notethatline5ofthefinalparagraphofthatremarkshouldread
‘are maximal finite cyclic subgroups’, and not ‘aremaximal finite subgroups’.
PROPOSITION 9. Let n ě 2, let x P ta,bu, and let y P tx,xu, where x “ an (resp. y “ bn´1) if
x “ a (resp. x “ b). Then ZBnpRP2qpyq “ xxy.
p p
Pproof. Let xp,x and y be as in the statempent. By [GG5, Corollary 4], ZPnpRP2qpxq “ xxy
is finite (the result also holds if n “ 2 since P pRP2q – Q ). It follows from the short
2 8
exact sequepnce (1) that ZBnpRP2qpxq is finite, and the inclusion ZBnpRP2qpxq Ă ZBnpRP2qpxq
then implies that ZBnpRP2qpyq is finite. Furthermore, ZpZBnpRP2qpyqq Ą xyy, so the order
of ZpZBnpRP2qpyqq is at least4. On the other hand, ZBnpRP2qpyq clearly conptains the cyclic
group xxy of order 4pn´kq, where k “ 0 (resp. k “ 1) if x “ a (resp. x “ b), which
using [GG6, Theorem 5] and the subgroup structure of dicyclic and the binary poly-
hedral gproups (for the latter, see [GG7, Proposition 85] forpexample) ispmaximal as a
finite cyclic group. Suppose that ZBnpRP2qpyq ‰ xxy. This subgroup structure implies
that ZBnpRP2qpyq would then be either dZicyclic or binary polyhedral, and so its centre
ZpZBnpRP2qpyqq would be isomorphic to 2. To seepthis, note that if G is a dicyclic (resp.
binarypolyhedral)groupthenitpossessesauniqueelementgoforder2thatiscentral.
If G – Q then clearly ZpGq – Z . So assume that G fl Q . Then G{xgy is dihedral of
8 2 8
order at least6 (resp. isomorphic to one of A , S and A ), and so its centre is trivial. It
4 4 5
Z
thusfollowsthatZpGq “ xgy,andsoZpGq – 2. Buty P ZpZBnpRP2qpyqqbyconstruction,
and taking G “ ZBnpRP2qpyq yields a contradiction since y is of order at least 4. We thus
conclude that ZBnpRP2qpxq “ xxyasrequired.
REMARK 10. In the proof of Proposition 9, we use [GG5, Corollary 4], whose proof de-
p p
pends (via [GG5, Proposition 21] and [GG5, Proposition 17] in the case n “ 3) upon
the characterisation of the conjugacy classes of elements of order 4 given in [GG2, Pro-
position 21(3)]. During the discussion leading to Theorem 6(d), we realised that the
statement of the latter result is not correct, and that there are in fact five conjugacy
classes of elements of order 4 in B pRP2q (which give rise to four conjugacy classes of
3
Z
subgroups isomorphic to ), and not three. This does not however affect the validity
4
of[GG5, Corollary 4]. Anerratum for[GG2, Proposition 21(3)]will appearelsewhere.
We then have the following result, which answers the question posed in [GG5, Re-
mark22].
PROPOSITION 11. If n ě 2 thenthe number of conjugacy classesin BnpRP2q of pure braidsof
order4 isequal to pn´2q!p2n´1q.
Proof. Let tγ ,...,γ u be a set of coset representatives of P pRP2q in B pRP2q, let w P
1 n! n n
P pRP2qbeoforder4,andletrwsdenotetheconjugacy classofw in P pRP2q. By[GG5,
n n
10
