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A short note on the Massicot-Wagner method

Arturo Rodríguez Fanlo

TL;DR

The note extends the Massicot–Wagner method to asymmetric group actions by incorporating a controlling set $\Gamma$, enabling an asymmetric version of the recursive MW argument. It proves that, under suitable conditions on $A,B$ and the interaction set $S(AB)$ (via $\Gamma$), $\Lambda^n$ has a locally compact (Lie) model, and it provides a recursive construction of a decreasing sequence of subsets to realize this model. The paper further develops definable and semipositively definable variants, introducing definable MW systems $\ell^{\mathrm{dg}}$ and $\ell^{\mathrm{dt}}$, and shows how definable local compact models can be obtained using $\Gamma^{00}$ and the logic topology in saturated contexts. Overall, it broadens the MW framework to group actions and definable settings, enhancing applicability to both dynamical and model-theoretic contexts and yielding canonical, definable Lie models for approximate subgroups.

Abstract

We provide a general abstract statement of the Massicot-Wagner method: our main result is an assymetric version (i.e. a version for group actions) of the recursive Massicot-Wagner argument.

A short note on the Massicot-Wagner method

TL;DR

The note extends the Massicot–Wagner method to asymmetric group actions by incorporating a controlling set , enabling an asymmetric version of the recursive MW argument. It proves that, under suitable conditions on and the interaction set (via ), has a locally compact (Lie) model, and it provides a recursive construction of a decreasing sequence of subsets to realize this model. The paper further develops definable and semipositively definable variants, introducing definable MW systems and , and shows how definable local compact models can be obtained using and the logic topology in saturated contexts. Overall, it broadens the MW framework to group actions and definable settings, enhancing applicability to both dynamical and model-theoretic contexts and yielding canonical, definable Lie models for approximate subgroups.

Abstract

We provide a general abstract statement of the Massicot-Wagner method: our main result is an assymetric version (i.e. a version for group actions) of the recursive Massicot-Wagner argument.

Paper Structure

This paper contains 4 sections, 13 theorems, 8 equations.

Table of Contents

  1. Preliminaries
  2. Massicot-Wagner systems
  3. Massicot-Wagner method
  4. Definable versions

Key Result

Lemma 1

Let $G$ be a group and let $\Lambda\subseteq G$ be an approximate subgroup. Let $(G,\mathcal{A},\mu)$ be a content spaceSee d:content. space invariant by left translations, with $\Lambda\in\mathcal{A}$ and $0<\mu(\Lambda)<\infty$. Then, for any $n\in\mathbb{N}$, there is an approximate subgroup $S\s

Theorems & Definitions (41)

  • Lemma 1: Basic Massicot-Wagner massicot2015approximate
  • Theorem 2: Recursive Massicot-Wagner massicot2015approximate
  • Lemma 3: Basic Massicot-Wagner hrushovski2022amenability
  • Theorem 4: Recursive Massicot-Wagner
  • Definition 5: Content
  • Definition 6: Mean
  • Definition 7: Approximate subgroups
  • Remark 8
  • Definition 9: Commensurator
  • Lemma 11
  • ...and 31 more