Universal differentiability sets in Carnot groups of arbitrarily high step

We show that any Carnot group G with sufficiently many deformable directions contains a measure zero set N such that every Lipschitz map f: G→ ℝ is differentiable at some point of N. We also prove that model filiform groups satisfy this condition, extending some previous results to a class of Carnot groups of arbitrarily high step. Essential to our work is the question of whether the existence of an (almost) maximal directional derivative Ef(x) in a Carnot group implies the differentiability of a Lipschitz map f at x. We show that such an implication is valid in model Filiform groups for directions that are outside a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two.