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We introduce a -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.
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Jonas Beyrer, Elia Fioravanti, Merlin Incerti-Medici, CAT(0) cube complexes are determined by their boundary cross ratio. Groups Geom. Dyn. 15 (2021), no. 1, pp. 313–333DOI 10.4171/GGD/599