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

TL;DR

This paper classifies virtually cyclic subgroups of sphere braid groups B_n(S^2), providing complete results for most cases and detailed subgroup characterizations, advancing understanding of braid group structures on the 2-sphere.

Contribution

It offers the first comprehensive classification of virtually cyclic subgroups of B_n(S^2), including new characterizations of centralisers, normalisers, and subgroup isomorphism classes.

Findings

Complete classification for odd n and large even n

Explicit open cases for small even n

New results on subgroup centralisers and normalisers

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.