作用在具有多个极大点稳定子直至共轭的树上的Pro-p群
Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation
摘要：
我们证明了一个生成的亲$p$组$G$作用于亲$p$树$T$分裂为一个自由的融合亲$p$产品或亲$p$ HNN扩展的边缘稳定。若$G$与多个点稳定子作用直至共轭，则证明了它是一个有限的pro-$p$群$(\cal G, Γ)$的基本pro-$p$群，其中边和点群分别是$T$的某些点和边的稳定子.如果边稳定子是顺循环的，我们给出了$Γ$上的一个界，它是由$G$的生成元的最小数目表示的。我们还给出了一个原$p$群$G$可达的一个条件，即它可用第一上同调$H^1(G, \mathbb{F}_p[[G]])$表示.
英文摘要：
We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, Γ)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Γ$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $G$ to be accessible in terms of the first cohomology $H^1(G, \mathbb{F}_p[[G]])$.
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