We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension.

中文翻译：

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幂零群的类平移动作

我们为幂等群上的类翻译动作提供了新的障碍。假设给定两个有限生成的无扭幂零群，它们具有相同的多项式增长程度，但非同构卡诺补全（渐近锥）。我们证明不存在从一组到另一组的单射 Lipschitz 函数。由此可见，任何一方都不能像翻译一样对另一方采取行动。由于 Lipschitz 注入不必是双 Lipschitz 嵌入，因此这是对 Pansu 在相同同质维度组的背景下的经典结果的加强。