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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

THE BOHR COMPACTIFICATION OF AN ARITHMETIC GROUP

Résumé

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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Dates et versions

hal-04073352 , version 1 (18-04-2023)

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  • HAL Id : hal-04073352 , version 1

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Bachir Bekka. THE BOHR COMPACTIFICATION OF AN ARITHMETIC GROUP. 2023. ⟨hal-04073352⟩
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