Coarse structures on locally compact abelian groups
Metric Geometry
2024-08-16 v1 Group Theory
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.
Cite
@article{arxiv.2408.07813,
title = {Coarse structures on locally compact abelian groups},
author = {Dmitri Shakhmatov and Takamitsu Yamauchi and Nicolò Zava},
journal= {arXiv preprint arXiv:2408.07813},
year = {2024}
}