A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913041798397952 |
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| author | Speight, Gareth Zimmerman, Scott |
| author_facet | Speight, Gareth Zimmerman, Scott |
| contents | We construct a Lipschitz curve in the free Carnot group of step 3 with 2 generators that meets every $C^{1}$ horizontal curve in a set of measure zero. This shows that the $C^{1}_{H}$-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely $C^1_H$ 1-unrectifiable. Hence 1-rectifiability in Carnot groups is wildly different to its counterpart in Euclidean spaces, wherein the Whitney Extension Theorem guarantees that Lipschitz rectifiability and $C^1$ rectifiability are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16618 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves Speight, Gareth Zimmerman, Scott Metric Geometry 53C17 (Primary), 58C25 (Secondary) We construct a Lipschitz curve in the free Carnot group of step 3 with 2 generators that meets every $C^{1}$ horizontal curve in a set of measure zero. This shows that the $C^{1}_{H}$-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely $C^1_H$ 1-unrectifiable. Hence 1-rectifiability in Carnot groups is wildly different to its counterpart in Euclidean spaces, wherein the Whitney Extension Theorem guarantees that Lipschitz rectifiability and $C^1$ rectifiability are equivalent. |
| title | A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves |
| topic | Metric Geometry 53C17 (Primary), 58C25 (Secondary) |
| url | https://arxiv.org/abs/2604.16618 |