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BOREL DENSITY FOR APPROXIMATE LATTICES

Part of:Probabilistic methods in group theoryLinear algebraic groups and related topicsLocally compact groups and their algebras

Published online by Cambridge University Press:  05 November 2019

MICHAEL BJÖRKLUNDOpen the ORCID record for MICHAEL BJÖRKLUND[Opens in a new window],
TOBIAS HARTNICKOpen the ORCID record for TOBIAS HARTNICK[Opens in a new window] and
THIERRY STULEMEIJER
MICHAEL BJÖRKLUND
Affiliation:
Department of Mathematics, Chalmers, Gothenburg, Sweden; micbjo@chalmers.se
TOBIAS HARTNICK
Affiliation:
Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Germany; tobias.hartnick@kit.edu
THIERRY STULEMEIJER
Affiliation:
Mathematisches Institut, Justus-Liebig-Universität Gießen, Germany; thierry.stulemeijer@math.uni-giessen.de

Abstract

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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 theory20G25: Linear algebraic groups over local fields and their integers

Information

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

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