The Bohr compactification of an arithmetic group
HTML articles powered by AMS MathViewer
- by Bachir Bekka;
- Trans. Amer. Math. Soc. 378 (2025), 3755-3777
- DOI: https://doi.org/10.1090/tran/9422
- Published electronically: March 19, 2025
- HTML | PDF | Request permission
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
- Hirotada Anzai and Shizuo Kakutani, Bohr compactifications of a locally compact Abelian group. I, Proc. Imp. Acad. Tokyo 19 (1943), 476–480. MR 15122
- H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
- Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
- B. Bekka and P. de la Harpe, Unitary representations of groups, duals, and characters, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2020., DOI 10.1090/surv/250
- Harald Bohr, Zur theorie der fast periodischen funktionen, Acta Math. 45 (1925), no. 1, 29–127 (German). I. Eine verallgemeinerung der theorie der fourierreihen. MR 1555192, DOI 10.1007/BF02395468
- Harald Bohr, Zur Theorie der Fastperiodischen Funktionen, Acta Math. 46 (1925), no. 1-2, 101–214 (German). II. Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen. MR 1555201, DOI 10.1007/BF02543859
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Armand Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30. MR 202718, DOI 10.1007/BF02684289
- Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712, DOI 10.1007/BF02684375
- Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
- N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. MR 358652
- Claude Chevalley, Deux théorèmes d’arithmétique, J. Math. Soc. Japan 3 (1951), 36–44 (French). MR 44570, DOI 10.2969/jmsj/00310036
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 458185
- Cornelia Druţu and Michael Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications, vol. 63, American Mathematical Society, Providence, RI, 2018. With an appendix by Bogdan Nica. MR 3753580, DOI 10.1090/coll/063
- John Franks and Michael Handel, Distortion elements in group actions on surfaces, Duke Math. J. 131 (2006), no. 3, 441–468. MR 2219247, DOI 10.1215/S0012-7094-06-13132-0
- S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125–158. MR 1114609, DOI 10.2307/2944334
- Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
- Joan E. Hart and Kenneth Kunen, Bohr compactifications of non-abelian groups, Proceedings of the 16th Summer Conference on General Topology and its Applications (New York), 2001/02, pp. 593–626. MR 2032839
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
- Per Holm, On the Bohr compactification, Math. Ann. 156 (1964), 34–46. MR 181700, DOI 10.1007/BF01359979
- Alexander Lubotzky, Shahar Mozes, and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 5–53 (2001). MR 1828742, DOI 10.1007/BF02698740
- G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
- G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221. MR 92928, DOI 10.2307/2372490
- J. v. Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. 36 (1934), no. 3, 445–492. MR 1501752, DOI 10.1090/S0002-9947-1934-1501752-3
- Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
- M. S. Raghunathan, On the congruence subgroup problem, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 107–161. MR 507030, DOI 10.1007/BF02684320
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 507234, DOI 10.1007/978-3-642-86426-1
- Luis Ribes and Pavel Zalesskii, Profinite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2000. MR 1775104, DOI 10.1007/978-3-662-04097-3
- I. E. Segal and John von Neumann, A theorem on unitary representations of semisimple Lie groups, Ann. of Math. (2) 52 (1950), 509–517. MR 37309, DOI 10.2307/1969429
- Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527 (French). MR 272790, DOI 10.2307/1970630
- J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
- Jacques Tits, Groupes à croissance polynomiale (d’après M. Gromov et al.), Bourbaki Seminar, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer, Berlin-New York, 1981, pp. 176–188 (French). MR 647496
- B. A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 76, Springer-Verlag, New York-Heidelberg, 1973. MR 335656
- André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 5741
- Dave Witte Morris, Introduction to arithmetic groups, Deductive Press, [place of publication not identified], 2015. MR 3307755
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