On metric approximate subgroups

TL;DR

This paper extends the concept of approximate subgroups to metric groups, proving that under certain conditions, such subsets are close to simpler approximate subgroups, linking metric and Lie group structures.

Contribution

It generalizes previous discrete group results to metric groups, establishing a connection between metric approximate subgroups and Lie groups under finite covering assumptions.

Findings

Metric approximate subgroups are close to simpler ones under certain conditions.

In groups with bounded exponent, approximate subgroups can be approximated by single-translate subsets.

The results bridge metric group theory and Lie group structure analysis.

Abstract

Let $G$ be a group with a metric $\mathrm{d}$ invariant under left and right translations, and let $\bar{\mathbb{D}}_r$ be the ball of radius $r$ around the identity. A $(k,r)$-metric approximate subgroup is a symmetric subset $X$ of $G$ such that the pairwise product set $XX$ is covered by at most $k$ translates of $X\bar{\mathbb{D}}_r$. This notion was introduced in arXiv:math/0601431 along with the version for discrete groups (approximate subgroups). In arXiv:0909.2190, it was shown for the discrete case that, at the asymptotic limit of $X$ finite but large, the "approximateness" (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on $X$ replacing finiteness. In particular, if $G$ has bounded exponent, we show that any $(k,r)$-metric approximate subgroup is close to a $(1,r')$-metric approximate subgroup for an appropriate $r'$.