#### Volume 8, issue 2 (2008)

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

### Daciberg L Gonçalves and John Guaschi

Algebraic & Geometric Topology 8 (2008) 757–785
##### 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 ${B}_{n}\left({\mathbb{S}}^{2}\right)$: ${ℤ}_{2\left(n-1\right)}$; 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 ${B}_{n}\left({\mathbb{S}}^{2}\right)$.

##### Keywords
braid group, configuration space, finite group, mapping class group, conjugacy class, lower central series, derived series
##### Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 20F50, 20E45, 57M99