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Closed approximate subgroups: compactness, amenability and approximate lattices

Simon Machado

TL;DR

The paper investigates closed approximate subgroups of locally compact groups, with a focus on approximate lattices and finite-co-volume phenomena. It introduces and elaborates the notion of good models, proving a Cartan-type closed-subgroup theorem for closed approximate subgroups and a structure theorem for amenable closed approximate subgroups in the spirit of Breuillard–Green–Tao, including a Meyer-type extension to amenable ambient groups. It then extends Meyer's model-set description to approximate lattices via good models, and proves Auslander–Mostow-type density and intersection results for approximate lattices, linking model-set containment to cut-and-project constructions. The methods leverage model-theoretic perspectives (saturated extensions) and universal properties of good models to derive universal and Bohr-type compactifications, with strong consequences for Euclidean and linear settings as well as Borel-density-type results. Collectively, the work provides a unifying framework for understanding approximate lattices in broad classes of groups, with applications to aperiodic order, harmonic analysis, and the geometry of locally compact groups.

Abstract

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices i.e. those approximate subgroups that are discrete and have finite co-volume. We prove an approximate subgroup version of Cartan's closed-subgroup theorem and study some applications. We give a structure theorem for closed approximate subgroups of amenable groups in the spirit of the Breuillard--Green--Tao theorem. We then prove two results concerning approximate lattices: we extend to amenable groups a structure theorem for mathematical quasi-crystals due to Meyer; we prove results concerning intersections of radicals of Lie groups and discrete approximate subgroups generalising theorems due to Auslander, Bieberbach and Mostow. As an underlying theme, we exploit the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. We show how one can draw information about a given approximate subgroup from a good model, when it exists.

Closed approximate subgroups: compactness, amenability and approximate lattices

TL;DR

The paper investigates closed approximate subgroups of locally compact groups, with a focus on approximate lattices and finite-co-volume phenomena. It introduces and elaborates the notion of good models, proving a Cartan-type closed-subgroup theorem for closed approximate subgroups and a structure theorem for amenable closed approximate subgroups in the spirit of Breuillard–Green–Tao, including a Meyer-type extension to amenable ambient groups. It then extends Meyer's model-set description to approximate lattices via good models, and proves Auslander–Mostow-type density and intersection results for approximate lattices, linking model-set containment to cut-and-project constructions. The methods leverage model-theoretic perspectives (saturated extensions) and universal properties of good models to derive universal and Bohr-type compactifications, with strong consequences for Euclidean and linear settings as well as Borel-density-type results. Collectively, the work provides a unifying framework for understanding approximate lattices in broad classes of groups, with applications to aperiodic order, harmonic analysis, and the geometry of locally compact groups.

Abstract

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices i.e. those approximate subgroups that are discrete and have finite co-volume. We prove an approximate subgroup version of Cartan's closed-subgroup theorem and study some applications. We give a structure theorem for closed approximate subgroups of amenable groups in the spirit of the Breuillard--Green--Tao theorem. We then prove two results concerning approximate lattices: we extend to amenable groups a structure theorem for mathematical quasi-crystals due to Meyer; we prove results concerning intersections of radicals of Lie groups and discrete approximate subgroups generalising theorems due to Auslander, Bieberbach and Mostow. As an underlying theme, we exploit the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. We show how one can draw information about a given approximate subgroup from a good model, when it exists.

Paper Structure

This paper contains 26 sections, 60 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on approximate subgroups and commensurability
  4. Strong approximate lattices and cut-and-project schemes
  5. Good models : definition, first properties and examples
  6. About the definition of good models
  7. Group-theoretic characterisation of good models
  8. Universal properties and compatibility with limits
  9. An approximate subgroup without a good model
  10. Good models and cut-and-project schemes
  11. A closed-approximate-subgroup theorem
  12. Globalisation in Hausdorff Topological Groups
  13. Closed approximate subgroups of Euclidean spaces
  14. Structure of Compact Approximate Subgroups
  15. Bohr-type Compactification
  16. ...and 11 more sections

Key Result

Proposition 1.1

Let $\Lambda$ be an approximate lattice in a locally compact second countable group $G$. Then $\Lambda$ is a Meyer subset if and only if it is contained in and commensurable to a model set.

Theorems & Definitions (126)

  • Definition 1
  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4: Closed-approximate-subgroup theorem
  • Proposition 1.5
  • Theorem 1.6: Structure of compact approximate subgroups
  • Theorem 1.7: Meyer theorem for amenable groups
  • Theorem 1.8: Structure of amenable approximate subgroups
  • Theorem 1.9
  • ...and 116 more