Hyperbolicity of the sub-Riemannian affine-additive group

TL;DR

This paper studies the geometric properties of the affine-additive group with a sub-Riemannian metric, showing it is hyperbolic and not quasiconformally equivalent to certain well-known groups, with implications for quasiregular maps.

Contribution

It establishes the hyperbolicity and Ahlfors regularity of the affine-additive group and demonstrates its distinct quasiconformal geometry compared to the Heisenberg and roto-translation groups.

Findings

The affine-additive group is locally 4-Ahlfors regular.

It is hyperbolic with non-vanishing 4-capacity at infinity.

Quasiregular maps from the Heisenberg group to the affine-additive group are constant.

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.