Thompson groups for systems of groups, and their finiteness properties
Thompson groups for systems of groups, and their finiteness properties
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群系统的汤普森群及其有限性
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. C. B. Zaremsky
中科院分区：
文献类型：
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作者：
Stefan Witzel;M. C. B. Zaremsky
我们描述了一个程序，用于构建一个广义汤普森组的一个家庭的群体，配备了我们所谓的克隆系统。先前已知的汤普森群F、V、Vbr和Fbr分别使用平凡群、对称群、辫子群和纯辫子群的系统从这个过程中产生。
我们给出了新的例子，家庭的群体，承认克隆系统和研究如何有限性所产生的广义汤普森群的性质取决于那些原来的群体。这里主要的新例子包括上三角矩阵群、模拟反射群和环辫群。对于整体函数域的S-整数环上的上三角矩阵群的广义Thompson群，给出了有限性性质的新的证明方法，证明了广义Thompson群的有限长恰是族中群的有限长的极限劣长.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups.
We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.