Stable group theory and approximate subgroups

TL;DR

This paper explores the structure of approximate subgroups in linear groups, connecting model theory, algebraic groups, and combinatorics to show that such groups are close to nilpotent or finite, with broad implications.

Contribution

It introduces new results linking stable model theory with finite combinatorics, demonstrating that approximate subgroups are near subgroups or algebraic subsets, and proves key theorems in a general first-order framework.

Findings

Approximate subgroups are close to actual subgroups or algebraic subsets.

Finitely generated groups with approximate subgroups are virtually nilpotent.

Established the independence and stabilizer theorems in a broad logical setting.

Abstract

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromov's proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.