Bornological Metrics on Groups

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.