Schlichting's Theorem for Approximate Subgroups

TL;DR

This paper extends Schlichting's theorem to approximate subgroups, showing that a uniform family of commensurable approximate subgroups admits an invariant approximate subgroup closely related to the original family.

Contribution

It proves Schlichting's theorem for approximate subgroups, establishing the existence of an invariant approximate subgroup for uniform families of commensurable approximate subgroups.

Findings

Existence of an invariant approximate subgroup for uniform families

Extension of Schlichting's theorem to approximate subgroups

Framework for analyzing approximate subgroups in group theory

Abstract

We prove Schlichting's theorem for approximate subgroups: if $\mathcal{X}$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with $\mathcal{X}$.