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Groups, Geometry, and Dynamics


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Volume 15, Issue 1, 2021, pp. 313–333
DOI: 10.4171/GGD/599

Published online: 2021-03-25

CAT(0) cube complexes are determined by their boundary cross ratio

Jonas Beyrer[1], Elia Fioravanti[2] and Merlin Incerti-Medici[3]

(1) Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France
(2) Universität Bonn, Germany
(3) Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France

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.

Keywords: CAT(0) cube complexes, cross ratios, Roller boundary

Beyrer Jonas, Fioravanti Elia, Incerti-Medici Merlin: CAT(0) cube complexes are determined by their boundary cross ratio. Groups Geom. Dyn. 15 (2021), 313-333. doi: 10.4171/GGD/599