The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Abstract

Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi's classification of the torsion elements of B_n(S^2), and explain how the finite subgroups of B_n(S^2) are related to this classification, as well as to the lower central and derived series of B_n(S^2).

arXiv comment23 pages

References in corpus (4)

• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• Configuration spaces of C and CP^1: some analytic properties
• Conjugacy classes of finite subgroups of certain mapping class groups
• The quaternion group as a subgroup of the sphere braid groups

Cited by in corpus (3)

• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• The classification of the virtually cyclic subgroups of the sphere braid groups
• Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces