Cross ratios on CAT(0) cube complexes and marked length-spectrum rigidity

We show that group actions on irreducible \CAT cube complexes with no free faces are uniquely determined by their \ell^1 length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of \emphnon-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of \CAT cube complexes. Inspired by results in \emphnegative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths. As a special case of our work, we construct a compactification of the Char-ney–Stambaugh–Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler–Vogtmann Outer Space.