The classification of the virtually cyclic subgroups of the sphere braid groups

Abstract

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete classification. For small even values of n, the classification is complete up to an explicit finite number of open cases. In order to prove our main theorem, we obtain a number of other results of independent interest, notably the characterisation of the centralisers and normalisers of the finite cyclic and dicyclic subgroups of B_n(S^2), a result concerning conjugate powers of finite order elements, an analysis of the isomorphism classes of the amalgamated products that occur as subgroups of B_n(S^2), as well as an alternative proof of the fact that the universal covering space of the n-th configuration space of S^2 has the homotopy type of S^3 if n is greater than or equal to three.

96 pages, 11 figures

References in corpus (4)

• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces
• The classification and the conjugacy classes of the finite subgroups of the sphere braid groups
• Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

Cited by in corpus (5)

• Cusps of hyperbolic 4-manifolds and rational homology spheres
• Free and properly discontinuous actions of groups on homotopy $2n$-spheres
• Mapping degrees between spherical $3$-manifolds
• The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n/[P_n,P_n]$
• On distinct finite covers of 3-manifolds