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${\rm CAT(0)}$ cube complexes are determined by their boundary cross ratio (1805.08478v4)
Published 22 May 2018 in math.GT and math.GR
Abstract: We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ 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.