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We study $K$-theoretical aspects of the braid groups $B\_n(\mathbb{S}^{2})$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~\cite{JM}.In light of this, in order to determine the algebraic $K$-theory of the groupring $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$, one should first compute that of itsvirtually cyclic subgroups, which were classified by D.~L.~Gon{\c c}alves andthe first author. We calculate the Whitehead and $K\_{-1}$-groups of the grouprings of the finite subgroups (dicyclic and binary polyhedral) of$B\_n(\mathbb{S}^{2})$ for all $4\leq n\leq 11$. Some new phenomena occur, suchas the appearance of torsion for the $K\_{-1}$-groups. We then go on to studythe case $n=4$ in detail, which is the smallest value of $n$ for which$B\_n(\mathbb{S}^{2})$ is infinite. We show that $B\_n(\mathbb{S}^{2})$ is anamalgamated product of two finite groups, from which we are able to determine auniversal space for proper actions of the group $B\_n(\mathbb{S}^{2})$. We alsocalculate the algebraic $K$-theory of the infinite virtually cyclic subgroupsof $B\_n(\mathbb{S}^{2})$, including the Nil groups of the quaternion group oforder $8$. This enables us to determine the lower algebraic $K$-theory of$\mathbb{Z}[B\_n(\mathbb{S}^{2})]$.

The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B\_4(\mathbb{S}^2)]$