Infinite Approximate Subgroups of Soluble Lie Groups

TL;DR

This paper investigates the structure of infinite approximate subgroups within soluble Lie groups, demonstrating their proximity to genuine subgroups and extending results on quasi-crystals to this context.

Contribution

It generalizes Fried and Goldman's theorem, providing a new structure theorem for approximate lattices in soluble Lie groups, extending Meyer's quasi-crystal theorem.

Findings

Approximate subgroups are close to genuine connected subgroups.

Established a structure theorem for approximate lattices in soluble Lie groups.

Extended Meyer's quasi-crystal theorem to soluble Lie groups.

Abstract

We study infinite approximate subgroups of soluble Lie groups. Generalising a theorem of Fried and Goldman we show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building up on this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.