An upper bound for the length of a traveling salesman path in the Heisenberg group

We show that a sufficient condition for a subset E in the Heisenberg group (endowed with the Carnot-Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of [6], allowing for any power r < 4 to replace the power 2 of the Jones-β-number. This complements in an open endedway ourwork in [13], where we showed that such an estimate was necessary, however with the power r = 4.