论文标题
右角Artin Pro- $ P $组
Right-angled Artin pro-$p$ groups
论文作者
论文摘要
令$ p $为素数。与Fnite Simplicial Graph $γ$相关的右角aRTIN PRO-P $ $ p $g_γ$是与$γ$相关的右角artin组的Pro-P $完成。我们证明以下断言是等效的：（i）$γ$的诱导子图是一个正方形或一条具有四个顶点的线（长度为3的路径）； （ii）$g_γ$的每个封闭子组本身都是一个直角的Artin Pro-P $组（可能是无限生成的）； （iii）$g_γ$是bloch-kato pro- $ p $ group; （iv）$g_γ$的每个封闭子组都有无扭转的Abelianization； （v）$g_γ$作为最大pro-p $ p $ galois $ g_k（p）$ g_k（p）$含有原始$ p $ p $ p $ th root of Unity; （vi）$g_γ$可以通过迭代两个组理论操作来从$ \ mathbb {z} _p $构建，即用$ \ mathbb {z} _p $和免费的pro-pro- $ p $ punders的直接产品。这解决了Quadrelli和Weigel的肯定猜想。 Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for right-angled Artin pro-$p$ groups.此外，我们证明$g_γ$在且只有大于三个的$γ$长度的每个电路都具有和弦。
Let $p$ be a prime. The right-angled Artin pro-$p$ group $G_Γ$ associated to a fnite simplicial graph $Γ$ is the pro-$p$ completion of the right-angled Artin group associated to $Γ$. We prove that the following assertions are equivalent: (i) no induced subgraph of $Γ$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of $G_Γ$ is itself a right-angled Artin pro-$p$ group (possibly infinitely generated); (iii) $G_Γ$ is a Bloch-Kato pro-$p$ group; (iv) every closed subgroup of $G_Γ$ has torsion free abelianization; (v) $G_Γ$ occurs as the maximal pro-$p$ Galois group $G_K(p)$ of some field $K$ containing a primitive $p$th root of unity; (vi) $G_Γ$ can be constructed from $\mathbb{Z}_p$ by iterating two group theoretic operations, namely, direct products with $\mathbb{Z}_p$ and free pro-$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for right-angled Artin pro-$p$ groups. Moreover, we prove that $G_Γ$ is coherent if and only if each circuit of $Γ$ of length greater than three has a chord.
