Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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The Bohr compactification of an arithmetic group
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by Bachir Bekka;
Trans. Amer. Math. Soc. 378 (2025), 3755-3777
DOI: https://doi.org/10.1090/tran/9422
Published electronically: March 19, 2025

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 $\mathbf {G}(\mathbf {R})$.
References
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Bibliographic Information
  • Bachir Bekka
  • Affiliation: Univ Rennes, CNRS, IRMAR–UMR 6625, Campus Beaulieu, F-35042 Rennes Cedex, France
  • MR Author ID: 33840
  • Email: bachir.bekka@univ-rennes1.fr
  • Received by editor(s): June 6, 2023
  • Received by editor(s) in revised form: November 5, 2023, November 29, 2023, and December 22, 2023
  • Published electronically: March 19, 2025
  • Additional Notes: The author was supported by the ANR (French Agence Nationale de la Recherche) through the project Labex Lebesgue (ANR-11-LABX-0020-01).
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 3755-3777
  • MSC (2020): Primary 22D10, 22C05, 20E18
  • DOI: https://doi.org/10.1090/tran/9422