Approximate lattices

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.