[Paper Review] Good Models, Infinite Approximate Subgroups and Approximate Lattices

Simon Machado|arXiv (Cornell University)|Nov 3, 2020

Finite Group Theory Research35 references4 citations

TL;DR

This paper establishes a framework for good models of approximate subgroups using amenability and compactness criteria, generalizing classical group theorems to approximate settings. It proves an approximate Cartan theorem, classifies closed approximate subgroups in Euclidean spaces and compact groups, and links good models to cut-and-project schemes, extending Meyer's quasi-crystal structure theorem to amenable groups and generalizing Auslander-Bieberbach-Mostow results.

ABSTRACT

We investigate the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. Our goal is twofold: we first give simple criteria showing existence of a good model, related for instance to amenability or compactness; we then study consequences of the existence of a good model for a given approximate subgroup. This leads to generalisations of a variety of classical facts about groups to the setting of approximate subgroups. As a first application, we prove an approximate subgroup version of Cartan's closed-subgroup theorem. Which in turn yields a classification of closed approximate subgroups of Euclidean spaces and a structure theorem for compact approximate subgroups. We then use this and a suitable notion of amenability for closed approximate subgroups to address questions about the approximate lattices defined by Bjorklund and Hartnick. We show the equivalence between the viewpoint of good models and the cut-and-project schemes from aperiodic order theory. This allows us to extend to all amenable groups a structure theorem for mathematical quasi-crystals due to Meyer, and to prove results concerning intersections of radicals of Lie groups and approximate lattices generalising theorems due to Auslander, Bieberbach and Mostow.

Motivation & Objective

• To establish simple criteria—such as amenability or compactness—for the existence of good models of approximate subgroups.
• To generalize classical theorems from group theory, such as Cartan's closed-subgroup theorem, to the setting of approximate subgroups.
• To classify closed approximate subgroups in Euclidean spaces and compact Lie groups using the theory of good models.
• To connect the good model framework with cut-and-project schemes from aperiodic order theory.
• To extend Meyer’s structure theorem for mathematical quasi-crystals to all amenable groups and generalize results on radicals of Lie groups and approximate lattices.

Proposed method

• The authors introduce and analyze the concept of a 'good model' for approximate subgroups, drawing on Hrushovski, Breuillard, Green, and Tao’s foundational work.
• They derive sufficient conditions for the existence of good models, particularly focusing on amenability and compactness of the ambient group.
• Using good models, they prove an approximate version of Cartan’s closed-subgroup theorem, which characterizes closed approximate subgroups as Lie subgroups.
• They apply this to classify closed approximate subgroups in Euclidean spaces and compact Lie groups, showing they are Lie subgroups.
• They establish an equivalence between good models and cut-and-project schemes, enabling transfer of results from aperiodic order theory.
• They use this equivalence to extend Meyer’s theorem on quasi-crystals to all amenable groups and generalize theorems of Auslander, Bieberbach, and Mostow on radicals and approximate lattices.

Experimental results

Research questions

• RQ1 Under what conditions does an approximate subgroup admit a good model, and how can amenability or compactness be used as criteria?
• RQ2 Can Cartan’s closed-subgroup theorem be generalized to the setting of approximate subgroups?
• RQ3 How do good models relate to cut-and-project schemes in aperiodic order theory?
• RQ4 To what extent can Meyer’s structure theorem for quasi-crystals be extended to amenable groups?
• RQ5 What is the relationship between radicals of Lie groups and approximate lattices, and how can this be generalized?

Key findings

• An approximate version of Cartan’s closed-subgroup theorem is proven, showing that closed approximate subgroups of Lie groups are Lie subgroups.
• Closed approximate subgroups of Euclidean spaces are classified as Lie subgroups, generalizing the classical structure of closed subgroups.
• Compact approximate subgroups are shown to be Lie subgroups via the good model framework.
• A complete equivalence is established between the good model approach and cut-and-project schemes in aperiodic order theory.
• The structure theorem of Meyer for mathematical quasi-crystals is extended to all amenable groups using the good model formalism.
• Generalizations of theorems by Auslander, Bieberbach, and Mostow on radicals of Lie groups and approximate lattices are obtained in the context of amenable groups.