New topological methods to solve equations over groups
New topological methods to solve equations over groups
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求解群方程的新拓扑方法
DOI：
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发表时间：
2015
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通讯作者：
Andreas Berthold Thom
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作者：
A. Klyachko;Andreas Berthold Thom
我们证明了与Gast {mathbf F}_2$中的群字$w相关联的方程可以在超线性群$G$上求解，如果它的内容--即它在${mathbf F}_2$中的增广--不位于${mathbf F}_2$的下中心列的第二项中。此外，如果$G$是有限的，那么可以在$G$的有限扩张中找到一个解。该方法的证明扩展技术开发的Gerstenhaber和罗索斯，并使用计算$p$-局部同伦理论和上同调紧李群。
We show that the equation associated with a group word $w in G ast {mathbf F}_2$ can be solved over a hyperlinear group $G$ if its content - that is its augmentation in ${mathbf F}_2$ - does not lie in the second term of the lower central series of ${mathbf F}_2$. Moreover, if $G$ is finite, then a solution can be found in a finite extension of $G$. The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in $p$-local homotopy theory and cohomology of compact Lie groups.