Infinite generation of the kernels of the Magnus and Burau representations
Infinite generation of the kernels of the Magnus and Burau representations
复制标题
Magnus 和 Burau 表示的内核的无限生成
DOI：
10.2140/agt.2010.10.837
复制
发表时间：
2009
影响因子：
0.7
通讯作者：
B. Farb
中科院分区：
文献类型：
--
作者：
Thomas Church;B. Farb
考虑Torelli群的Magnus表示的核Mag_g和辫子群的Burau表示的核Bur_n。证明了当g >= 2，n >= 6时，群Mag和Bur n有无穷秩一阶同调.因此，我们得出结论，没有任何组有任何有限的生成集。在每种情况下的证明方法包括生产一种“约翰逊型”同态无限秩阿贝尔群，并证明图像有无限秩。对于Bur_n的情形，我们借助于计算机计算来完成.
Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of "Johnson-type" homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of Bur_n, we do this with the assistance of a computer calculation.