The lower central and derived series of the braid groups of the sphere and the punctured sphere
| dc.creator | Gonçalves, Daciberg Lima | |
| dc.creator | Guaschi, John | |
| dc.date | 2006-03-30 | |
| dc.date.accessioned | 2026-07-07T13:08:07Z | |
| dc.date.available | 2026-07-07T13:08:07Z | |
| dc.description | Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its quotients. For n \geq 1, the class of m-string braid groups B\_m(S^2) \ {x\_1,...,x\_n} of the n-punctured sphere includes the Artin braid groups B\_m, those of the annulus, and certain Artin and affine Artin groups. We extend results of Gorin and Lin, and show that the LCS (resp. DS) of B\_m is determined for all m (resp. for all m\neq 4). For m=4, we obtain some elements of the DS. When n\geq 2, we prove that the LCS (resp. DS) of B\_m(S^2) \ {x\_1,...,x\_n} is constant from the commutator subgroup onwards for all m\geq 3 (resp. m\geq 5). We then show that B\_2(S^2\{x\_1,x\_2}) is residually nilpotent, that its LCS coincides with that of Z\_2*Z, and that the Γ\_i/Γ\_{i+1} are 2-elementary finitely-generated groups. For m\geq 3 and n=2, we obtain a presentation of the derived subgroup and its Abelianisation. For n=3, we see that the quotients Γ\_i/Γ\_{i+1} are 2-elementary finitely-generated groups. | |
| dc.description | 103 pages | |
| dc.identifier | https://arxiv.org/abs/math/0603701 | |
| dc.identifier | http://arxiv.org/abs/math/0603701 | |
| dc.identifier | doi:10.1090/S0002-9947-09-04766-7 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/228316 | |
| dc.subject | Geometric Topology | |
| dc.subject | Group Theory | |
| dc.subject | Primary: 20F36, 20F14. Secondary: 20F05, 55R80, 20E26 | |
| dc.title | The lower central and derived series of the braid groups of the sphere and the punctured sphere | |
| dc.type | text |