  In the setting of Carnot groups, we exhibit examples of intrinisc Lipschitz
curves of positive $\mathcal{H}^1$-measure that intersect every connected
intrinsic Lipschitz curve in a $\mathcal{H}^1$-negligible set. As a consequence
such curves cannot be extended to connected intrinsic Lipschitz curves.
  The examples are constructed in the Engel group and in the free Carnot group
of step 3 and rank 2. While the failure of the Lipschitz extension property was
already known for some pairs of Carnot groups, ours is the first example of the
analogous phenomenon for intrinsic Lipschitz graphs. This is in sharp contrast
with the Euclidean case.

Metric Geometry