Bornological Metrics on Groups

TL;DR

This paper introduces and studies bornological metrics on countable groups, characterizing their coarse equivalence classes, and constructs examples of improper metrics that differ from proper ones.

Contribution

It defines bornological metrics, links coarse classes to bornologies, and provides criteria for metrizability and invariance, with applications to improper metrics.

Findings

Each coarse class of bornological metrics is determined by a bornology.

Every coarse class has a canonical left-invariant representative.

Constructs improper metrics on finitely generated groups that are non-equivalent to proper metrics.

Abstract

Let $G$ be a countable group. We study left-invariant metrics on $G$ that are not necessarily proper, introducing the notion of a \emph{bornological metric}: a metric $\rho$ such that for every $C>0$ there exists $S_C>0$ with the property that $\rho(x,y)<C$ implies $\rho(gx,gy)<S_C$ for all $g\in G$. We show that each coarse equivalence class of bornological metrics is determined by a bornology on $G$, and that every such class contains a canonical left-invariant representative. The metrizability of a bornology is characterized in terms of countable generation of the associated coarse structure, and a criterion for strong $G$-invariance of a coarse structure is established. As an application, we construct families of improper left-invariant metrics on finitely generated groups that are pairwise non-equivalent and not coarsely equivalent to any proper left-invariant metric.