论文标题
有限生成的nilpotent组的一致性亚组问题
The Congruence Subgroup Problem for finitely generated Nilpotent Groups
论文作者
论文摘要
有限生成的组$γ$和$ g \ leq aut（γ）$的一致性子组问题询问地图$ \ hat {g} \ to aut（\hatγ）$是注入性的，还是更一般而言，其内核$ c \ left是什么（g，γ\ \ right）$？这里$ \ hat {x} $表示$ x $的涂料完成。在情况下，$ g = aut（γ）$我们表示$ c \ left（γ\右）= c \ left（aut（γ），γ\右）$。 令$γ$为有限生成的组，$ \barγ=γ/[γ，γ] $和$γ^{*} = \barγ/tor（\barγ）\ cong \ mathbb {z}^{（d）} $。表示$ aut^{*}（γ）= \ textrm {im}（aut（γ）\ to aut（γ^{*}）））\ leq gl_ {d}（\ mathbb {z}）$。在本文中，我们表明，当$γ$ nilpotent时，有一个规范的同构$ c \ left（γ\ right）\ simeq c（aut^{*}（γ）（γ），γ^{*}）$。换句话说，$ c \ left（γ\右）$完全取决于算术组$ aut^{*}（γ）$的经典一致性子组问题的解决方案。 特别是，如果$γ=ψ_{n，c} $是$ n $元素上的$ c $ class $ c $的免费生成的，我们得到$ c（ψ_{n，c}）= c（\ mathbb {z} $ c（ψ_{2，c}）= c（\ mathbb {z}^{（2）}）= \ hat {f}_Ω$ =可计数的发电机数量上的免费profinite组。
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphism $C\left(Γ\right)\simeq C(Aut^{*}(Γ),Γ^{*})$. In other words, $C\left(Γ\right)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Aut^{*}(Γ)$. In particular, in the case where $Γ=Ψ_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Ψ_{n,c})=C(\mathbb{Z}^{(n)})=\{e\}$ whenever $n\geq3$, and $C(Ψ_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_ω$ = the free profinite group on countable number of generators.
