球面辫群的虚循环子群的低代数$K$ -理论和$\mathbb{Z}[B\_4(\mathbb{S}^2)]$
The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B\_4(\mathbb{S}^2)]$
摘要：
本文研究了$2$ -球面上$n$弦上的辫群$B\_n(\mathbb{S}^{2})$的$K$ -理论方面的问题，根据后两位作者的结果，已知它们满足Farrell-Jones同构猜想~ \cite{JM} .鉴于此，为了确定群环$\mathbb{Z}[B\_n(\mathbb{S}^{2})]$的代数$K$ -理论，首先要计算它的虚循环子群的代数$K$-理论，D.~ L.~阿尔维斯和第一作者。我们计算了$B\_n(\mathbb{S}^{2})$的有限子群（双循环和二元多面体）的群环的Whitehead和$K\_{-1}$ -群，对所有$4\leq n\leq 11$。出现了一些新的现象，如$K\_{-1}$ -群出现了挠.然后，我们继续详细研究$n=4$的情况，这是$n$的最小值，其中$B\_n(\mathbb{S}^{2})$是无限的。证明了$B\_n(\mathbb{S}^{2})$是两个有限群的合并积，由此我们能够确定群$B\_n(\mathbb{S}^{2})$的真作用的泛空间.我们还计算了$B\_n(\mathbb{S}^{2})$的无限虚循环子群的代数$K$ -理论，包括$8$阶四元数群的Nil群。这使我们能够确定较低的代数$K$ -理论的$\mathbb{Z}[B\_n(\mathbb{S}^{2})]$。
英文摘要：
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 group ring $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon{\c c}alves and the first author. We calculate the Whitehead and $K\_{-1}$-groups of the group rings 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, 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 $B\_n(\mathbb{S}^{2})$ is infinite. We show that $B\_n(\mathbb{S}^{2})$ is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group $B\_n(\mathbb{S}^{2})$. We also calculate the algebraic $K$-theory of the infinite virtually cyclic subgroups of $B\_n(\mathbb{S}^{2})$, including the Nil groups of the quaternion group of order $8$. This enables us to determine the lower algebraic $K$-theory of $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$.
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