We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of CAT(0) cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.

中文翻译：

---

CAT（0）立方体复合物由其边界交叉比确定

我们在CAT（0）多维数据集的Roller边界上引入了$ \ mathbb {Z} $值的交叉比率。我们通过证明每个保留Roller边界的跨比率双向射影唯一地扩展为立方同构来激发其相关性。我们的结果惊人地通用，甚至适用于具有平凡自同构群的无限维，局部无限立方复合体。