arXiv 0810.20272008-10math.GR

非负亏投射有限群的正规子群

Normal Subgroups of Profinite Groups of Non-negative Deficiency

Fritz Grunewald, Andrei Jaikin-Zapirain, Aline G. S. Pinto, Pavel A. Zalesski

备注：: This is a revised version; we added a result stating that arithmetic Klenian groups are good

摘要：

本文首次研究了非负亏的投射有限群。本文的主要目的是证明在非负亏的profinite群$G$中存在无穷指数的n-生成正规子群对$G$的结构给出了相当强的推论.为了使这一点更精确，我们引入了profinite群$G$的$p$ -亏格（$p$是素数）的概念。这个概念在研究投射有限群时比亏数的概念更有用。证明了若$G$的$p$ -亏是正的，且$N$是一个可生成的正规子群，使得$G/N$的$p$ -Sylow子群是无穷的，且$p$整除$N$的阶，则上同调$p$ -维数有$\cd_p(G)=2$，$\cd_p(N)=1$和$\vcd_p(G/N)=1$;此外，或者$G/N$的$p$-Sylow子群实际上是环状的，或者$N$的$p$-Sylow子群是环状的。一个在素数$p$（$PD^3$ -群）上的3维Profinite Poincaré对偶群$G$有亏数0。在这种情况下，我们表明，对于上述$N$和$p$，或者$N$在$p$处是$PD^1$，而$G/N$在$p$处实际上是$PD^2$，或者$N$在$p$处是$PD^2$，而$G/N$在$p$处实际上是$PD^1$。特别地，如果$G$是pro-$p$，则$N$是无限循环的并且$G/N$实际上是Demushkin，或者$N$是Demushkin并且$G/N$实际上是无限循环的。我们应用这一结果推导出结构信息的profinite完成的上升HNN-扩展的自由群。我们也给出了我们的理论的某些算术群的同余核的一些影响。

英文摘要：

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ divides the order of $N$ then we have $\cd_p(G)=2$, $\cd_p(N)=1$ and $\vcd_p(G/N)=1$ for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of $G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. A profinite Poincaré duality group $G$ of dimension 3 at a prime $p$ ($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ as above either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$ is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ is pro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or $N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups. We also give some implications of our theory to the congruence kernels of certain arithmetic groups.

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摘要：

英文摘要：

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