We study K-theoretical aspects of the braid groups B n pS 2 q on n strings of the 2-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture [56]. In light of this, in order to determine the algebraic K-theory of the group ring ZrB n pS 2 qs, one should first compute that of its virtually cyclic subgroups, which were classified by D. L. Gonçalves and the first author [47]. We calculate the Whitehead and K´1-groups of the group rings of the finite subgroups (dicyclic and binary polyhedral) of B n pS 2 q for all 4 ď n ď 11. Some new phenomena occur, such as the appearance of torsion for the K´1-groups. We then go on to study the case n " 4 in detail, which is the smallest value of n for which B n pS 2 q is infinite. We show that B 4 pS 2 q is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group B 4 pS 2 q. We also calculate the algebraic K-theory of the infinite virtually cyclic subgroups of B 4 pS 2 q, including the Nil groups of the quaternion group of order 8. This enables us to determine the lower algebraic K-theory of ZrB 4 pS 2 qs. arXiv:1209.4791v2 [math.KT]
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