Coarse structures on locally compact abelian groups

TL;DR

This paper explores the large-scale geometry of locally compact abelian groups by analyzing group ideals, establishing a key equivalence between relative compactness and coarse boundedness, and demonstrating embedding limitations for Banach spaces.

Contribution

It introduces a comparison of group ideals in locally compact groups and proves a novel equivalence between relative compactness and coarse boundedness in abelian groups.

Findings

A subset of a locally compact abelian group is relatively compact iff it is coarsely bounded.

Infinite-dimensional Banach spaces cannot embed into products of locally compact groups.

Abstract

Motivated by the study of the large-scale geometry of topological groups, we investigate particular families of subsets of topological groups named group ideals. We compare different group ideals in the realm of locally compact groups. In particular, we show that a subset of a locally compact abelian group is relatively compact if and only if it is coarsely bounded. Using this result, we prove that an infinite-dimensional Banach space cannot be embedded into any product of locally compact groups.