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THE BOHR COMPACTIFICATION OF AN ARITHMETIC GROUP

Authors
  • Bekka, Bachir
Publication Date
Apr 18, 2023
Source
HAL
Keywords
Language
English
License
Unknown
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Abstract

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/[Γ,Γ])_0 × Prof(Γ). In the general case, using a Levi decomposition G = U ⋊ 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 G(R).

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