  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.
