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Hyperbolicity of the sub-Riemannian affine-additive group

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arxiv 2407.04635 v1 pith:6SAS4IMY submitted 2024-07-05 math.MG math.DG

Zoltán M. Balogh , Elia Bubani , Ioannis D. Platis This is my paper · ORCID
classification math.MGmath.DG
keywords groupaffine-additivemeasuremetricheisenbergleft-invariantspacesub-riemannian
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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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Cited by 1 Pith paper

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  1. Stretch maps on the affine-additive group

    math.DG 2024-11 conditional novelty 6.0 of 10

    Linear and radial stretch maps on the affine-additive group are proved to minimize the mean quasiconformal distortion functional within classes of maps with prescribed boundary behavior.

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