Hyperbolicity of the sub-Riemannian affine-additive group

• DOI: 10.48550/arxiv.2407.04635
• Publication Date: 2024-01-01
• Authors: Balogh, Zoltán M.; Bubani, Elia; Platis, Ioannis D.
• Keywords: Mathematics - Differential Geometry; Mathematics - Metric Geometry; Differential Geometry (math.DG); 53C17, 30L10; FOS: Mathematics; Metric Geometry (math.MG)

17 pages<br/>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.<br/>