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

Abstract

Let $n\ge 3$. We classify the finite groups which are realised as subgroups of the sphere braid group $B_n\left({\mathbb{S}}^2\right)$. 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 $B_n\left({\mathbb{S}}^2\right)$: ${\mathbb{Z}}_{2\left(n-1\right)}$; the dicyclic groups of order $4n$ and $4\left(n-2\right)$; the binary tetrahedral group $T^{\ast }$; the binary octahedral group $O^{\ast }$; and the binary icosahedral group $I^{\ast }$. We give geometric as well as some explicit algebraic constructions of these groups in $B_n\left({\mathbb{S}}^2\right)$ and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi’s classification of the torsion elements of $B_n\left({\mathbb{S}}^2\right)$ and explain how the finite subgroups of $B_n\left({\mathbb{S}}^2\right)$ are related to this classification, as well as to the lower central and derived series of $B_n\left({\mathbb{S}}^2\right)$.