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15 October 2018 Approximate lattices
Michael Björklund, Tobias Hartnick
Duke Math. J. 167(15): 2903-2964 (15 October 2018). DOI: 10.1215/00127094-2018-0028

Abstract

In this article we introduce and study uniform and nonuniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasicrystals (Meyer sets) in lcsc Abelian groups. We show that envelopes of strong approximate lattices are unimodular and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor–Schwarz lemma for uniform approximate lattices in compactly generated lcsc groups, which we then use to relate the metric amenability of uniform approximate lattices to the amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of irreducible higher-rank symmetric spaces of noncompact type are quasi-isometrically rigid with respect to finitely generated approximate groups.

Citation

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Michael Björklund. Tobias Hartnick. "Approximate lattices." Duke Math. J. 167 (15) 2903 - 2964, 15 October 2018. https://doi.org/10.1215/00127094-2018-0028

Information

Received: 1 February 2017; Revised: 21 May 2018; Published: 15 October 2018
First available in Project Euclid: 3 October 2018

zbMATH: 06982210
MathSciNet: MR3865655
Digital Object Identifier: 10.1215/00127094-2018-0028

Subjects:
Primary: 20N99
Secondary: 20F65 , 22F10

Keywords: approximate groups , Delone sets in groups , quasi-isometric rigidity

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 15 • 15 October 2018
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