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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.04635 |
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| _version_ | 1866911945837248512 |
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| author | Balogh, Zoltán M. Bubani, Elia Platis, Ioannis D. |
| author_facet | Balogh, Zoltán M. Bubani, Elia Platis, Ioannis D. |
| contents | We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity of the sub-Riemannian affine-additive group Balogh, Zoltán M. Bubani, Elia Platis, Ioannis D. Metric Geometry Differential Geometry 53C17, 30L10 We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant. |
| title | Hyperbolicity of the sub-Riemannian affine-additive group |
| topic | Metric Geometry Differential Geometry 53C17, 30L10 |
| url | https://arxiv.org/abs/2407.04635 |