Approximate lattices

TL;DR

This paper introduces and studies approximate lattices in locally compact second countable groups, generalizing classical lattices and quasi-crystals, and explores their properties, envelopes, and rigidity phenomena.

Contribution

It defines uniform and non-uniform approximate lattices, proves their envelopes are unimodular, and establishes key properties and rigidity results extending classical theorems.

Findings

Envelopes of strong approximate lattices are unimodular.

Approximate lattices in nilpotent groups are uniform.

Higher rank symmetric spaces have isometry groups that are QI rigid with respect to approximate groups.

Abstract

In this article we introduce and study uniform and non-uniform 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 quasi-crystals (a.k.a. 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 metric amenability of uniform approximate lattices to amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of higher rank symmetric spaces of non-compact type are QI rigid with respect to finitely-generated approximate groups.