Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T04:45:06.305Z Has data issue: false hasContentIssue false
  • English
  • Français

BOREL DENSITY FOR APPROXIMATE LATTICES

Part of: Probabilistic methods in group theory Linear algebraic groups and related topics Locally compact groups and their algebras

Published online by Cambridge University Press:  05 November 2019

MICHAEL BJÖRKLUND Open the ORCID record for MICHAEL BJÖRKLUND [Opens in a new window] ,
TOBIAS HARTNICK Open the ORCID record for TOBIAS HARTNICK [Opens in a new window]  and
THIERRY STULEMEIJER
MICHAEL BJÖRKLUND
Affiliation:
Department of Mathematics, Chalmers, Gothenburg, Sweden; [email protected]
TOBIAS HARTNICK
Affiliation:
Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Germany; [email protected]
THIERRY STULEMEIJER
Affiliation:
Mathematisches Institut, Justus-Liebig-Universität Gießen, Germany; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We extend classical density theorems of Borel and Dani–Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.

MSC classification

Primary: 22D40: Ergodic theory on groups
Secondary: 20P05: Probabilistic methods in group theory 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e40
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Bader, U., Shalom, Y. and Yehuda, ‘Factor and normal subgroup theorems for lattices in products of groups’, Invent. Math. 163(2) (2006), 415–454.Google Scholar
Björklund, M. and Hartnick, T., ‘Approximate lattices’, Duke Math. J. 167(15) (2018), 2903–2964.Google Scholar
Björklund, M. and Hartnick, T., ‘Analytic properties of approximate lattices’. Ann. Inst. Fourier (Grenoble), to appear, Preprint, 2017, arXiv:1709.09942.Google Scholar
Björklund, M. and Hartnick, T., ‘Spectral theory of approximate lattices in nilpotent Lie groups’, Preprint, 2018, arXiv:1811.06563.Google Scholar
Björklund, M., Hartnick, T. and Pogorzelski, F., ‘Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets’, Proc. Lond. Math. Soc. 116(4) (2018), 957–996.Google Scholar
Borel, A., ‘Density properties for certain subgroups of semi-simple groups without compact components’, Ann. of Math. (2) 72(1) (1960), 179–188.Google Scholar
Cornulier, Y. and de la Harpe, P., Metric Geometry of Locally Compact Groups, Tracts in Mathematics, 25 (EMS Publishing House, Zürich, 2016).Google Scholar
Dani, S. G., ‘On ergodic quasi-invariant measures of group automorphism’, Israel J. Math 43(1) (1982), 62–74.Google Scholar
Furstenberg, H., ‘A note on Borel’s density theorem’, Proc. Amer. Math. Soc. 55(1) (1976), 209–212.Google Scholar
Furstenberg, H. and Glasner, E., ‘Stationary dynamical systems’, inDynamical Numbers—Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics, 532 (American Mathematical Society, Providence, RI) 1–28.Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, 52 (Springer, New York - Heidelberg, 1977).Google Scholar
Machado, S., ‘Approximate lattices and Meyer sets in nilpotent Lie groups’, Preprint, 2018, arXiv:1810.10870.Google Scholar
Margulis, G. A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17 (Springer, Berlin, 1991).Google Scholar
Meyer, Y., Algebraic Numbers and Harmonic Analysis, North-Holland Mathematical Library, 2 (North-Holland Publishing Co., New York, 1972).Google Scholar
Milne, J. S., Algebraic Groups, Cambridge Studies in Advanced Mathematics 170 (Cambridge University Press, Cambridge, 2017).Google Scholar
Oesterlé, J., ‘Nombres de Tamagawa et groupes unipotents en caractéristique p ’, Invent. Math. 78(1) (1984), 13–88.Google Scholar
Paulin, F., ‘De la geometrie et de la dynamique de SLn(ℝ) et SLn(ℤ)’, inSur la dynamique des groupes de matrices et applications arithmetiques, Editions de l’Ecole Polytechnique (eds. Berline, N., Plagne, A. and Sabbah, C.) (Editions Ellipses, Paris, 2007), 47–110.Google Scholar
Shalom, Y., ‘Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)’, Trans. Amer. Math. Soc. 351(8) (1999), 3387–3412.Google Scholar
Stuck, G., ‘Growth of homogeneous spaces, density of discrete subgroups and Kazhdan’s property (T)’, Invent. Math. 109(3) (1992), 505–517.Google Scholar
Tao, T., ‘Product set estimates for non-commutative groups’, Combinatorica 28(5) (2008), 547–594.Google Scholar
Tits, J., Oeuvres/Collected Works, Vol. IV, (European Mathematical Society, Zürich, 2013).Google Scholar