Rough Approximate subgroups

TL;DR

The paper extends Hrushovski's Lie Model framework to rough approximate subgroups by introducing $T$-rough measures and a robust thick-set machinery. By defining $T$-rough definable amenability via a sequence $(\mu_i)$ of left-translation-invariant, finite-approximate measures, it proves that under a finiteness condition there exists a type-definable normal subgroup $N$ of bounded index with $T_\omega\subseteq N\subseteq A^4T_\omega$, yielding a Lie-model description. In metric-group contexts, the results specialize through ultraproducts to give a connected Lie model $\pi: H\to L$ with kernel controlled by $T_\omega$ and $A^4T_\omega$, and, when the roughness constant is uniform, to $A^2$ being a $T_\omega$-rough $K$-approximate subgroup; these recover and strengthen HR22 for metric groups. Overall, the work provides a language-free, subadditivity-based path to Lie-model results for rough approximate subgroups with broad model-theoretic and geometric implications.

Abstract

Given a $T$-rough definably amenable $T$-rough approximate subgroup $A$ of a group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ of bounded index in $\langle A\rangle$.

Theorems & Definitions (31)

• Theorem 1
• Corollary 2
• Definition 3
• Lemma 4
• proof
• Remark 5
• Definition 6
• Lemma 7
• proof
• Lemma 8