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

Daciberg Lima Gonçalves (1) , John Guaschi (2)

1 IME - Instituto de Matemática e Estatística
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; 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).

Mots clés: sphere braid groups; finite subgroups; mapping class group; conjugacy classes; periodic cohomology

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Daciberg Lima Gonçalves, John Guaschi. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebraic and Geometric Topology, 2008, 8 (2), pp.757-785. ⟨10.2140/agt.2008.8.757⟩. ⟨hal-00191422⟩