The Lower Algebraic K-Theory of Virtually Cyclic Subgroups of the Braid Groups of the Sphere and of ZB4(S2) , 1st ed. 2018 SpringerBriefs in Mathematics Series

This volume deals with the K-theoretical aspects of the group rings of braid groups of the 2-sphere. The lower algebraic K-theory of the finite subgroups of these groups up to eleven strings is computed using a wide variety of tools. Many of the techniques extend to the general case, and the results reveal new K-theoretical phenomena with respect to the previous study of other families of groups. The second part of the manuscript focusses on the case of the 4-string braid group of the 2-sphere, which is shown to be hyperbolic in the sense of Gromov. This permits the computation of the infinite maximal virtually cyclic subgroups of this group and their conjugacy classes, and applying the fact that this group satisfies the Fibred Isomorphism Conjecture of Farrell and Jones, leads to an explicit calculation of its lower K-theory. 

Introduction.- Lower algebraic K-theory of the finite subgroups of Bn(S²).- The braid group B4(S²) and the conjugacy classes of its maximal virtually cyclic subgroups.- Lower algebraic K-theory groups of the group ring Z[B4(S²)].- Appendix A: The fibred isomorphism conjecture.- Appendix B: Braid groups.- References.

Includes many worked examples of K-theory computations for finite groups

A useful reference for researchers in K-theory, bringing together a broad array of techniques and references in one place, and mainly self-contained

Applies the knowledge of virtually-cyclic subgroups to determine the lower algebraic K-theory for the braid groups of B4(S2)

Gives new properties about braid groups of the sphere