The lower central and derived series of the braid groups of the sphere and the punctured sphere

Details The lower central and derived series of the braid groups of the sphere and the punctured sphere The lower central and derived series of the braid groups of the sphere and the punctured sphere

2006-03-30

arxiv.org/abs/math/0603701v1

Mathematics

Geometric Topology

103 pages

Scientific paper

10.1090/S0002-9947-09-04766-7

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 \Gamma\_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 \Gamma\_i/\Gamma\_{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 \Gamma\_i/\Gamma\_{i+1} are 2-elementary finitely-generated groups.

Affiliated with

Also associated with

No associations

Rating

The lower central and derived series of the braid groups of the sphere and the punctured sphere does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The lower central and derived series of the braid groups of the sphere and the punctured sphere, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The lower central and derived series of the braid groups of the sphere and the punctured sphere will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-141720