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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2601.07702 |
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| _version_ | 1866911369690873856 |
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| author | Grützner, Georg |
| author_facet | Grützner, Georg |
| contents | We introduce asymptotic-Möbius (AM) maps, a large-scale analogue of quasi-Möbius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits, and rigid enough to yield structural consequences absent from Pansu's notion of large-scale conformality. We establish basic properties of AM-maps, give several sources of examples, including quasi-isometries, sublinear bi-Lipschitz equivalences, snowflaking, and Assouad embeddings, and apply the theory to large-scale dimension and metric cotype. As applications we obtain dimension-monotonicity results for nilpotent groups and CAT(0) spaces, and new obstructions to the existence of AM-maps arising from metric cotype. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07702 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic-Möbius maps Grützner, Georg Metric Geometry 30L10 We introduce asymptotic-Möbius (AM) maps, a large-scale analogue of quasi-Möbius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits, and rigid enough to yield structural consequences absent from Pansu's notion of large-scale conformality. We establish basic properties of AM-maps, give several sources of examples, including quasi-isometries, sublinear bi-Lipschitz equivalences, snowflaking, and Assouad embeddings, and apply the theory to large-scale dimension and metric cotype. As applications we obtain dimension-monotonicity results for nilpotent groups and CAT(0) spaces, and new obstructions to the existence of AM-maps arising from metric cotype. |
| title | Asymptotic-Möbius maps |
| topic | Metric Geometry 30L10 |
| url | https://arxiv.org/abs/2601.07702 |