In this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$ of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$ has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$.

中文翻译：

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三阶自由群的IA-自同构群的低维上同调群

本文研究了IA-自同构群的有理上同调群的结构$\mathrm {IA}_3$利用组合群论和表示论对三阶自由群进行了求解。特别是，我们在第二个上同调群中检测到一个重要的不可约分量$\mathrm {IA}_3$，不包含在第一个上同调组的杯子产品图的图像中。我们还表明，第一个上同调群的三杯乘积是微不足道的。作为推论，我们得到下中心级数的第四项$\mathrm {IA}_3$在 Andreadakis-Johnson 过滤中具有有限指数$\mathrm {IA}_3$.