Abstract
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.
Highlights
Gromov boundaries of CAT(−1) spaces are naturally endowed with a notion of cross ratio
A classical example is provided by the standard projective cross ratio on ∂∞H2 ≃ RP1
We introduce a similar object on the Roller boundary of any CAT(0) cube complex X and show that this suffices to fully reconstruct the structure of X
Summary
Gromov boundaries of CAT(−1) spaces are naturally endowed with a notion of cross ratio. It is natural to consider the horoboundary of the metric space (X, d), usually known as Roller boundary ∂X This has become a standard tool in the study of cube complexes; see e.g. In [BF19b, BF19a], the Main Theorem is extended to cross-ratio preserving bijections between much smaller subsets of the Roller boundaries. This has applications to length-spectrum rigidity questions for actions on cube complexes. We conclude the introduction by remarking that the Main Theorem does not generalise to cross-ratio preserving embeddings ∂X ֒→ ∂Y This stands in contrast with the behaviour of trees [BS17] and rank-one symmetric spaces [Bou96]. ∂X ≃ ∂Y ֒→ ∂S, which does not extend to an isometric embedding X ֒→ S
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