Bornological Metrics on Groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911672102289408 |
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| author | Arutyunov, Andronick Perelygin, Artem |
| author_facet | Arutyunov, Andronick Perelygin, Artem |
| contents | 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 $ρ$ such that for every $C>0$ there exists $S_C>0$ with the property that $ρ(x,y)<C$ implies $ρ(gx,gy)<S_C$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bornological Metrics on Groups Arutyunov, Andronick Perelygin, Artem Group Theory Geometric Topology 20F65, 51F30, 54E35, 46A17 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 $ρ$ such that for every $C>0$ there exists $S_C>0$ with the property that $ρ(x,y)<C$ implies $ρ(gx,gy)<S_C$ 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. |
| title | Bornological Metrics on Groups |
| topic | Group Theory Geometric Topology 20F65, 51F30, 54E35, 46A17 |
| url | https://arxiv.org/abs/2605.10997 |