Hyperbolicity of the sub-Riemannian affine-additive group
Metric Geometry
2024-07-08 v1 Differential Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.2407.04635,
title = {Hyperbolicity of the sub-Riemannian affine-additive group},
author = {Zoltán M. Balogh and Elia Bubani and Ioannis D. Platis},
journal= {arXiv preprint arXiv:2407.04635},
year = {2024}
}
Comments
17 pages