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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

Let n 3. We classify the finite groups which are realised as subgroups of the sphere braid group Bn(S2). 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 Bn(S2): 2(n1); the dicyclic groups of order 4n 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 Bn(S2) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi’s classification of the torsion elements of Bn(S2) and explain how the finite subgroups of Bn(S2) are related to this classification, as well as to the lower central and derived series of Bn(S2).

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
Received: 26 November 2007
Revised: 11 February 2008
Accepted: 20 February 2008
Published: 25 May 2008
Daciberg L Gonçalves
Departamento de Matemática - IME-USP
Caixa Postal 66281 - Ag. Cidade de São Paulo
CEP: 05314-970 - São Paulo - SP
John Guaschi
Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139
Université de Caen
BP 5186
14032 Caen Cedex