Approximate groups, II: the solvable linear case

TL;DR

This paper analyzes the structure of approximate subgroups within solvable subgroups of GL_n(C), revealing they contain large nilpotent components and can be controlled by nilpotent progressions, advancing understanding of their algebraic structure.

Contribution

It characterizes approximate subgroups of solvable linear groups, demonstrating their containment of large nilpotent parts and their control by nilpotent progressions, extending previous results.

Findings

Approximate subgroups have a large nilpotent component.

They are efficiently controlled by nilpotent progressions.

The results connect solvable linear groups with nilpotent structures.

Abstract

We describe the structure of "K-approximate subgroups'' of solvable subgroups of GL_n(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free nilpotent groups, we show that such approximate subgroups are efficiently controlled by nilpotent progressions.