The Bohr compactification of an arithmetic group

Given a group Γ \Gamma , its Bohr compactification B o h r ( Γ ) Bohr(\Gamma ) and its profinite completion P r o f ( Γ ) Prof(\Gamma ) are compact groups naturally associated to Γ \Gamma ; moreover, P r o f ( Γ ) Prof(\Gamma ) can be identified with the quotient of B o h r ( Γ ) Bohr(\Gamma ) by its connected component B o h r ( Γ ) 0 Bohr(\Gamma )_0 . We study the structure of B o h r ( Γ ) Bohr(\Gamma ) for an arithmetic subgroup Γ \Gamma of an algebraic group G \mathbf {G} over Q \mathbf {Q} . When G \mathbf {G} is unipotent, we show that B o h r ( Γ ) Bohr(\Gamma ) can be identified with the direct product B o h r ( Γ A b ) 0 × P r o f ( Γ ) Bohr(\Gamma ^{\mathrm {Ab}})_0\times Prof(\Gamma ) , where Γ A b = Γ / [ Γ , Γ ] \Gamma ^{\mathrm {Ab}}= \Gamma /[\Gamma , \Gamma ] is the abelianization of Γ \Gamma . In the general case, using a Levi decomposition G = U ⋊ H \mathbf {G}= \mathbf {U}\rtimes \mathbf {H} (where U \mathbf {U} is unipotent and H \mathbf {H} is reductive), we show that B o h r ( Γ ) Bohr(\Gamma ) can be described as the semi-direct product of a certain quotient of B o h r ( Γ ∩ U ) Bohr(\Gamma \cap \mathbf {U}) with B o h r ( Γ ∩ H ) Bohr(\Gamma \cap \mathbf {H}) . When G \mathbf {G} is simple and has higher R \mathbf {R} -rank, B o h r ( Γ ) Bohr(\Gamma ) is isomorphic, up to a finite group, to the product K × P r o f ( Γ ) K\times Prof(\Gamma ) , where K K is the maximal compact factor of G ( R ) \mathbf {G}(\mathbf {R}) .