The classification and the conjugacy classes of the finite subgroups of the sphere braid groups
Abstract
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 T1; the binary octahedral group O1; 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).
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.