Approximate lattices
Artikel i vetenskaplig tidskrift, 2018

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 non-compact type are quasi-isometrically rigid with respect to finitely generated approximate groups.


Michael Björklund

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Tobias Hartnick

Justus-Liebig-Universität Gießen

Duke Mathematical Journal

0012-7094 (ISSN)

Vol. 167 15 2903-2964


Algebra och logik


Matematisk analys


Grundläggande vetenskaper



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