We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. 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.

中文翻译：

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任意高阶卡诺群中的普遍微分集

我们证明每个模型丝状群 $\mathbb{E}_{n}$ 都包含一个测度零集 $N$，使得每个 Lipschitz 映射 $f\colon \mathbb{E}_{n}\to \mathbb{R }$ 在 $N$ 的某个点是可微的。模型丝状群是一类可以具有任意高步长的卡诺群。我们工作的关键问题是，卡诺群中（几乎）最大方向导数 $Ef(x)$ 的存在是否意味着 Lipschitz 映射 $f$ 在 $x$ 处的可微性。我们表明，除了水平方向的一维子空间外，这种含义在模型丝状群中是有效的。相反，我们证明了这个蕴涵在第三步和第二阶的自由卡诺群中的每个水平方向都失败了。