We show that group actions on irreducible $\mathrm{CAT}(0)$ cube complexes with no free faces are uniquely determined by their ${\ell }^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are ${\ell }^1$-minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces).

中文翻译：

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CAT(0) 立方复合物的交叉比和显着的长谱刚性

我们证明了对不可约的群体行动$猫(0)$没有自由面的立方体复合体由它们的唯一决定${\ell }^1$长度函数。允许动作是非适当的和非协同的，只要它们是${\ell }^1$-minimal 并且在视觉边界上没有有限的轨道。这是，据我们所知，第一个长度谱刚性导致非正曲率设置（除了维数 2 和对称空间中的一些特殊情况）。