Duke Mathematical Journal

Approximate lattices

Michael Björklund and Tobias Hartnick

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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.

Article information

Duke Math. J., Volume 167, Number 15 (2018), 2903-2964.

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

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Zentralblatt MATH identifier

Primary: 20N99: None of the above, but in this section
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx]

approximate groups Delone sets in groups quasi-isometric rigidity


Björklund, Michael; Hartnick, Tobias. Approximate lattices. Duke Math. J. 167 (2018), no. 15, 2903--2964. doi:10.1215/00127094-2018-0028. https://projecteuclid.org/euclid.dmj/1538532050

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