当前位置: X-MOL 学术Groups Geom. Dyn. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The congruence subgroup problem for the free metabelian group on $n\geq4$ generators
Groups, Geometry, and Dynamics ( IF 0.6 ) Pub Date : 2020-08-13 , DOI: 10.4171/ggd/561
David El-Chai Ben-Ezra 1
Affiliation  

The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether the map $\widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma})$ is injective, or more generally, what is its kernel $C(\Gamma)$? Here $\widehat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C(\mathbb{Z}^{n})=\{ 1\}$ for every $n\geq3$, but $C(\mathbb{Z}^{2})=\widehat{F}_{\omega}$, where $\widehat{F}_{\omega}$ is the free profinite group on countably many generators.

Considering$\Phi_{n}$, the free metabelian group on $n$ generators, it was also proven that $C(\Phi_{2})=\widehat{F}_{\omega}$ and $C(\Phi_{3})\supseteq\widehat{F}_{\omega}$. In this paper we prove that $C(\Phi_{n})$ for $n\geq4$ is abelian. So, while the dichotomy in the abelian case is between $n=2$ and $n\geq3$, in the metabelian case it is between $n=2,3$ and $n\geq4$.



中文翻译:

$ n \ geq4 $生成器上的自由变位群的同余子群问题

有限生成组$ \ Gamma $的同余子组问题询问映射$ \ widehat {\ mathrm {Aut}(\ Gamma)} \到\ mathrm {Aut}(\ widehat {\ Gamma})$的映射是内射的,或更笼统地说,它的内核$ C(\ Gamma)$是什么?这里$ \ widehat {X} $表示$ X $的无限完成。众所周知,对于有限生成的自由阿贝尔群,每个$ n \ geq3 $$ C(\ mathbb {Z} ^ {n})= \ {1 \} $,但是$ C(\ mathbb {Z} ^ { 2})= \ widehat {F} _ {\ omega} $,其中$ \ widehat {F} _ {\ omega} $是许多生成器上的自由有限群。

考虑到$ \ Phi_ {n} $,即$ n $生成器上的自由变位群,还证明了$ C(\ Phi_ {2})= \ widehat {F} _ {\ omega} $和$ C(\ Phi_ {3})\ supseteq \ widehat {F} _ {\ omega} $。在本文中,我们证明$ n \ geq4 $的$ C(\ Phi_ {n})$是阿贝尔文。因此,尽管在阿贝利亚案例中的二分法在$ n = 2 $和$ n \ geq3 $之间,但在亚贝贝利亚案例中的二分法在$ n = 2,3 $和$ n \ geq4 $之间。

更新日期:2020-08-13
down
wechat
bug