The Bohr compactification of an arithmetic group

Abstract

Given a group $\Gamma ,$ its Bohr compactification $\mathrm{Bohr}(\Gamma )$ and its profinite completion $\mathrm{Prof}(\Gamma )$ are compact groups naturally associated to $\Gamma$; moreover, $\mathrm{Prof}(\Gamma )$ can be identified with the quotient of $\mathrm{Bohr}(\Gamma )$ by its connected component $\mathrm{Bohr}(\Gamma {)}_0.$ We study the structure of $\mathrm{Bohr}(\Gamma )$ for an arithmetic subgroup $\Gamma$ of an algebraic group $G$ over $\mathbf{Q}$. When $G$ is unipotent, we show that $\mathrm{Bohr}(\Gamma )$ can be identified with the direct product $\mathrm{Bohr}({\Gamma }^{\mathrm{Ab}}{)}_0\times \mathrm{Prof}(\Gamma )$, where ${\Gamma }^{\mathrm{Ab}}=\Gamma /[\Gamma ,\Gamma ]$ is the abelianization of $\Gamma .$ In the general case, using a Levi decomposition $G=U⋊H$ (where $U$ is unipotent and $H$ is reductive), we show that $\mathrm{Bohr}(\Gamma )$ can be described as the semi-direct product of a certain quotient of $\mathrm{Bohr}(\Gamma \cap U)$ with $\mathrm{Bohr}(\Gamma \cap H)$. When $G$ is simple and has higher $\mathbf{R}$-rank, $\mathrm{Bohr}(\Gamma )$ is isomorphic, up to a finite group, to the product $K\times \mathrm{Prof}(\Gamma ),$ where $K$ is the maximal compact factor of the real Lie group $G(\mathbf{R}).$

27 pages