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

Abstract

We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of $\mathrm{CAT}(0)$ 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.

arXiv comment19 pages, 3 figures. Final version, to appear on Groups Geom Dyn

References in corpus (6)

• Cubulating hyperbolic free-by-cyclic groups: the general case
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Cubulating hyperbolic free-by-cyclic groups: the irreducible case
• Cross ratios and cubulations of hyperbolic groups
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Cited by in corpus (5)

• Cross ratios and cubulations of hyperbolic groups
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Automorphisms of contact graphs of $\mathrm{CAT}(0)$ cube complexes
• Homotopy equivalent boundaries of cube complexes
• Möbius structures, quasi-metrics, and completeness