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

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 $4 n$ and $4 (n - 2)$ ; the binary tetrahedral group $T^{*}$ ; the binary octahedral group $O^{*}$ ; 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 $B_{n} (S^{2})$ .

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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

References

Publication

Received: 26 November 2007

Revised: 11 February 2008

Accepted: 20 February 2008

Published: 25 May 2008

Authors

Daciberg L Gonçalves
Departamento de Matemática - IME-USP Caixa Postal 66281 - Ag. Cidade de São Paulo CEP: 05314-970 - São Paulo - SP Brazil

John Guaschi
Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139 Université de Caen BP 5186 14032 Caen Cedex France

Volume 8, issue 2 (2008)

Daciberg L Gonçalves and John Guaschi

Abstract