  We show that the Heisenberg group is not minimal in looking down. This
answers Problem 11.15 in `Fractured fractals and broken dreams' by David and
Semmes, or equivalently, Question 22 and hence also Question 24 in
`Thirty-three yes or no questions about mappings, measures, and metrics' by
Heinonen and Semmes.
  The non-minimality of the Heisenberg group is shown by giving an example of
an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that
no Lipschitz map from a subset of $X$ to the Heisenberg group has image with
positive measure, and by providing a Lipschitz map from the Heisenberg group to
the space $X$ having as image the whole $X$.
  As part of proving the above result we define a new distance on the
Heisenberg group that is bounded by the Carnot-Carath\'eodory distance, that
preserves the Ahlfors-regularity, and such that the Carnot-Carath\'eodory
distance and the new distance are biLipschitz equivalent on no set of positive
measure. This construction works more generally in any Ahlfors-regular metric
space where one can make suitable shortcuts. Such spaces include for example
all snowflaked Ahlfors-regular metric spaces. With the same techniques we also
provide an example of a left-invariant distance on the Heisenberg group
biLipschitz to the Carnot-Carath\'eodory distance for which no blow-up admits
nontrivial dilations.
