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arXiv · 2407.04635

Hyperbolicity of the sub-Riemannian affine-additive group

Zoltán M. Balogh·Elia Bubani·Ioannis D. Platis
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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.

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Zoltán M. Balogh, Elia Bubani, Ioannis D. Platis. 2024-07-05. Hyperbolicity of the sub-Riemannian affine-additive group. https://arxiv.org/abs/2407.04635

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