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arXiv · 1805.08478

${\rm CAT(0)}$ cube complexes are determined by their boundary cross ratio

Jonas Beyrer·Elia Fioravanti·Merlin Incerti-Medici
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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.

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Jonas Beyrer, Elia Fioravanti, Merlin Incerti-Medici. 2019-11-08. ${\rm CAT(0)}$ cube complexes are determined by their boundary cross ratio. https://doi.org/10.4171/ggd%2F599

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