2023/04/18 by Bekka, Bachir
#20E18 #22C05 #22D10 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2304.09045
Given a group Γ, its Bohr compactification Bohr(Γ) and its profinite completion Prof(Γ) are compact groups naturally associated to Γ; moreover, Prof(Γ) can be identified with the quotient of Bohr(Γ) by its connected component Bohr(Γ)0. We study the structure of Bohr(Γ) for an arithmetic subgroup Γ of an algebraic group G over Q. When G is unipotent, we show that Bohr(Γ) can be identified with the direct product Bohr(Γ\rm Ab)0× 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 Bohr(Γ) can be described as the semi-direct product of a certain quotient of Bohr(Γ∩ U) with Bohr(Γ∩ H). When G is simple and has higher R-rank, Bohr(Γ) is isomorphic, up to a finite group, to the product K× Prof(Γ), where K is the maximal compact factor of the real Lie group G(R).