The Bohr compactification of an arithmetic group

Abstract:

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