Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

We show that every model filiform group ${\mathbb{E}}_n$ contains a measure zero set $N$ such that every Lipschitz map $f\colon $\mathbb{E}$_{n}\to $\mathbb{R}$$isdifferentiableatsomepointof$N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for 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.