Recursive Massicot–Wagner Argument

• The Recursive Massicot–Wagner Argument is a combinatorial and measure-theoretic method that recursively builds descending chains of symmetric approximate subgroups to model locally compact groups.
• It leverages invariant means, finite covering properties, and recursive inclusion to ensure each successive subgroup retains structural compatibility with the original set.
• This technique has broad applications in additive combinatorics, model theory, and ergodic theory, offering insights into amenability and definability within group actions.

The Recursive Massicot–Wagner Argument is a combinatorial and measure-theoretic technique, principally applied to the structural understanding of approximate subgroups and, in more recent developments, group actions on measure spaces. It establishes the existence of a descending chain of symmetric subsets within a group or its subgroup such that the chain encodes the passage to a locally compact (often Lie) group model, even in asymmetric situations involving group actions on external sets. The argument leverages properties of invariant means, finite covering, and recursive inclusion within powers of approximate subgroups or related definable sets.

1. Foundations: The Standard (Symmetric) Massicot–Wagner Method

The classical Massicot–Wagner argument investigates approximate subgroups $\Lambda$ of a group $G$, seeking a locally compact model through recursive descent. The basic lemma asserts that for any $n\in\mathbb{N}$, there exists a symmetric approximate subgroup $S\subseteq\Lambda^4$ satisfying $S^n\subseteq\Lambda^4$ (Fanlo, 22 Sep 2025). This forms the foundation for the recursive stepwise construction, where each stage produces a definable approximate subgroup $D_i$ (with $D_0=\Lambda$) such that $D_{i+1}^2\subseteq D_i$, and each $D_i$ remains commensurable with $\Lambda$ or a fixed power $\Lambda^m$. Compactness and quotient arguments guarantee that the existence of such a chain implies the presence of a type-definable normal subgroup of bounded index, permitting the quotient of $G$ by this subgroup to inherit a locally compact group structure.

2. Abstract Formulation and the Asymmetric Argument for Group Actions

Extending beyond the symmetric case, the argument accommodates arbitrary group actions on external sets $E$. The main result in (Fanlo, 22 Sep 2025) introduces an asymmetric version: given a group $G$ acting on $E$, and sets $A,B$ in an invariant mean space $(E,\mathcal{F},m)$, the key condition is that

$$S(AB) = \{g\in\langle\Lambda\rangle : m(gAB\cap AB)>0\} \subseteq \Lambda^m.$$

Here, $\Lambda$ is an approximate subgroup, and $S(AB)$ measures the "overlap" induced by the action. Instead of revising $A$ and $B$ at each step, the recursive process holds $A,B$ fixed and modifies control sets, such as $\Lambda$ and an auxiliary set $\Gamma\subseteq G$, to establish the requisite descending chain $D_n$ with $D_{n+1}^2\subseteq D_n$ and $D_n\subseteq\Lambda^m$ for all $n$. Thus, the existence of this sequence in the group action context is shown to imply a locally compact model for $\Lambda^m$.

3. Methodological Steps and Formulae

The recursive descent proceeds by:

• Starting with an initial approximate subgroup $\Lambda$ and mean $0<\mu(\Lambda)<\infty$.
• Applying the covering property: for any $W$ in a certain level $\ell_k$ of the Massicot–Wagner system,

$\{g\in\Gamma : gW\cap W\in\ell_{k-1}\}$

is "thick" in $\Lambda$, meaning there exists a finite subset $\Delta$ such that

$$\Lambda\subseteq\Delta\cdot\{g\in\Gamma : gW\cap W\in\ell_{k-1}\}.$$

• Recursively constructing a descending sequence $D_0=D, D_1,\dots$ with $D_{n+1}^2\subseteq D_n$ and $D_n\subseteq\Lambda^m$.
• Concluding with $K=\bigcap_n D_n$ and forming the quotient $\langle\Lambda\rangle/K$, which is locally compact.

This is accompanied by quantitative control: using parameters $\varepsilon$, $\lambda$ to force inequalities such as

$m(gWB\cap WB)>(1-\varepsilon)m(WB),$

with appropriate choice of $W\subseteq A$.

4. Extensions: Application to Amenability and Definability

The extension and refinement in (Hrushovski et al., 2019) demonstrate the argument's adaptability to settings involving amenable topological or definable groups. Here, stabilizer theorems leverage invariant means on lattices of $\bigvee$-definable sets, not just finitely supported ones, constructing generic, symmetric sets $Y$ such that

$Y^N \subseteq \mathrm{St}_1(BA)$

for a stabilizer $$\mathrm{St}_1(BA)=\{g\in G : m(g\cdot (BA)\cap BA)\geq (1-\varepsilon)m(BA)\}$$ and some $N$. This recursive approach enables the proof of equalities like $G^{00}=G^{000}$ connecting group-theoretic connected components and compactifications under amenability conditions. The recursive mechanism is essential for producing type-definable components which control the structure of approximate subgroups and their quotients.

5. Algebraic Analogues: Recursive Arguments via Binomial Recursions

Distinct but related methodologies appear in algebraic recursions, notably the "Recursive Massicot–Wagner Argument" in the context of generalized binomial coefficients and Jarden’s Theorem (Lang et al., 2013). Here, recursive strategies permit the derivation of recurrence relations for powers and products of recursive functions. Instead of relying on ratios that may be undefined when terms vanish, the argument defines coefficients intrinsically via polynomials, enabling a passage to the limit and unifying recurrence proofs in regular and degenerate cases:

$$\sum_{i=0}^{n+1}(-1)^i q^{i(i-1)/2} (n+1|i)_u X(m-i)=0,$$

where $(n+1|i)_u$ is a generalized binomial coefficient defined via polynomial evaluation at roots of the characteristic polynomial.

6. Connections to Additive Combinatorics and Model Theory

The method informs qualitative and quantitative structural results in additive combinatorics, such as the classification of approximate groups and their locally compact models central to results by Breuillard–Green–Tao and others. The recursive nature of the descending sequence construction provides concrete control over the approximate subgroup, ultimately linked to definability properties in model-theoretic settings. The argument’s utility extends to contexts including ergodic theory, topological dynamics, and the paper of non-abelian groups acting on auxiliary structures.

7. Concluding Remarks and Broader Significance

The Recursive Massicot–Wagner Argument provides a combinatorially transparent, measure-theoretically robust approach to the extraction of locally compact models for approximate subgroups, with a powerful generalization to group actions. It unifies former combinatoric and model-theoretic techniques—leveraging recursive chains of symmetric subsets and invariant means—to yield strong structural results on subgroups, their quotients, and compactifications. The asymmetric version further broadens its applicability, encapsulating contexts where group actions transcend left multiplication and intersect with combinatorial or topological properties of associated measure spaces. As such, the method occupies a central role in contemporary group theory, model theory, and additive combinatorics, as demonstrated in the foundational and recent literature (Fanlo, 22 Sep 2025, Hrushovski et al., 2019, Lang et al., 2013).