Cross ratios on cube complexes and length-spectrum rigidity

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).