Cross ratios naturally arise on boundaries of negatively curved spaces

and are a valuable tool in their study. If one however slightly relaxes the

curvature assumption, simply requiring it to be *non-positive*, things

tend to get more complicated. Even the mere definition of a cross ratio becomes a

more delicate matter.

Restricting to the context of CAT(0) cube complexes $X$, we observe that

most issues disappear if one considers the $\ell^1$ metric on $X$, rather than the

CAT(0) metric. We obtain a canonical cross ratio on the horoboundary of the $\ell^1$

metric, usually known as Roller boundary. This allows us to develop a general framework relating cross-ratio-preserving boundary maps to the study of length-spectrum rigidity for (not necessarily compact) cube complexes.

As an application, we show that essential, non-elementary actions on

irreducible CAT(0) cube complexes with no free faces are completely determined by

their marked $\ell^1$-length spectrum. One might wish to relax the no-free-faces

assumption and this is indeed possible for cubulations of hyperbolic groups, where

essentiality and hyperplane-essentiality actually suffice (and are necessary). We also

show that such cubulations of hyperbolic groups inject into the space of invariant

cross ratios on the Gromov boundary that are continuous at a co-meagre subset.

Joint work with J. Beyrer (Heidelberg) and M. Incerti-Medici (UZH).

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