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

Gonçalves, Daciberg Lima¹ ; Guaschi, John²

¹Departamento de Matemática - IME-USP, Caixa Postal 66281 - Ag. Cidade de São Paulo, CEP: 05314-970 - São Paulo - SP, Brazil
²Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen, BP 5186, 14032 Caen Cedex, France

Algebraic and Geometric Topology, Tome 8 (2008) no. 2, pp. 757-785

Résumé

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): ℤ2(n−1); 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).

Détail

DOI : 10.2140/agt.2008.8.757

Keywords: braid group, configuration space, finite group, mapping class group, conjugacy class, lower central series, derived series

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