算术群的玻尔紧化
The Bohr compactification of an arithmetic group
AI总结：
本文研究算术群的玻尔紧化结构，通过阿贝尔化、列维分解及高R-秩单群情形，将Bohr(Γ)分解为紧李因子与投射有限完备化的乘积或半直积形式。
摘要：
给定一个群Γ，其玻尔紧化Bohr(Γ)和投射有限完备化Prof(Γ)是与Γ自然关联的紧群；此外，Prof(Γ)可等同于Bohr(Γ)除以其连通分支Bohr(Γ)₀所得的商。我们研究定义在Q上的代数群G的算术子群Γ的Bohr(Γ)结构。当G为幂单群时，我们证明Bohr(Γ)可等同于直积Bohr(Γ^Ab)₀×Prof(Γ)，其中Γ^Ab=Γ/[Γ,Γ]是Γ的阿贝尔化。在一般情形下，利用列维分解G=U⋊H（其中U为幂单群，H为约化群），我们证明Bohr(Γ)可描述为Bohr(Γ∩U)的某个商与Bohr(Γ∩H)的半直积。当G为单群且具有较高R-秩时，Bohr(Γ)在相差一个有限群的范围内同构于乘积K×Prof(Γ)，其中K是实李群G(R)的极大紧因子。
英文摘要：
Given a group $Γ,$ its Bohr compactification $\operatorname{Bohr}(Γ)$ and its profinite completion $\operatorname{Prof}(Γ)$ are compact groups naturally associated to $Γ$; moreover, $\operatorname{Prof}(Γ)$ can be identified with the quotient of $\operatorname{Bohr}(Γ)$ by its connected component $\operatorname{Bohr}(Γ)_0.$ We study the structure of $\operatorname{Bohr}(Γ)$ for an arithmetic subgroup $Γ$ of an algebraic group $G$ over $\mathbf{Q}$. When $G$ is unipotent, we show that $\operatorname{Bohr}(Γ)$ can be identified with the direct product $\operatorname{Bohr}(Γ^{\rm Ab})_0\times \operatorname{Prof}(Γ)$, where $Γ^{\rm Ab}= Γ/[Γ, Γ]$ is the abelianization of $Γ.$ In the general case, using a Levi decomposition $G= U\rtimes H$ (where $U$ is unipotent and $H$ is reductive), we show that $\operatorname{Bohr}(Γ)$ can be described as the semi-direct product of a certain quotient of $\operatorname{Bohr}(Γ\cap U)$ with $\operatorname{Bohr}(Γ\cap H)$. When $G$ is simple and has higher $\mathbf{R}$-rank, $\operatorname{Bohr}(Γ)$ is isomorphic, up to a finite group, to the product $K\times \operatorname{Prof}(Γ),$ where $K$ is the maximal compact factor of the real Lie group $G(\mathbf{R}).$
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