BiLipschitz decomposition of Lipschitz maps between Carnot groups

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Let f: G → H be a Lipschitz map between two Carnot groups. We show that if B is ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that B Z can be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of meyerson, which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not admitting such a discretization.