The lower central and derived series of the braid groups of the projective plane


Autoria(s): GONCALVES, Daciberg Lima; GUASCHI, John
Contribuinte(s)

UNIVERSIDADE DE SÃO PAULO

Data(s)

20/10/2012

20/10/2012

2011

Resumo

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

international Cooperation USP/Cofecub project

international Cooperation USP/Cofecub project[105/06]

CNRS/CNPq[21119]

Centre National de la Recherche Scientifique (CNRS)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

ANR

ANR[ANR-08-BLAN-0269-02]

Identificador

JOURNAL OF ALGEBRA, v.331, n.1, p.96-129, 2011

0021-8693

http://producao.usp.br/handle/BDPI/30690

10.1016/j.jalgebra.2010.12.007

http://dx.doi.org/10.1016/j.jalgebra.2010.12.007

Idioma(s)

eng

Publicador

ACADEMIC PRESS INC ELSEVIER SCIENCE

Relação

Journal of Algebra

Direitos

restrictedAccess

Copyright ACADEMIC PRESS INC ELSEVIER SCIENCE

Palavras-Chave #Surface braid group #Projective plane braid group #Lower central series #Derived series #Configuration space #Exact sequence #SHORT EXACT SEQUENCE #CONFIGURATION-SPACES #SURFACE #SUBGROUPS #SPHERE #Mathematics
Tipo

article

original article

publishedVersion