Cross ratios on cube complexes and length-spectrum rigidity

We prove two versions of the marked length-spectrum rigidity conjecture for a large class of non-positively curved spaces: CAT(0) cube complexes. Under weak assumptions, we show that proper cocompact actions of Gromov hyperbolic groups on CAT(0) cube complexes are fully determined by their combinatorial length functions. If the cube complexes are irreducible and have the geodesic extension property, the same result holds for non-proper, non-cocompact actions of arbitrary groups. While some of our arguments are well-known in negative curvature, until now no length-spectrum rigidity result had been obtained in a setting of non-positive curvature (except for some particular cases in dimension 2). Our spaces are allowed to have arbitrarily high dimension and can contain flats. Fixing a group Γ, our work sets the basis for the study of the space of all actions of Γ on CAT(0) cube complexes. As a first step in this direction, we use length functions to compactify large spaces of cubulations of Γ. This can be viewed as an analogue of Thurston’s compactification of Teichmüller space. As a special case of our results, we construct a compactification of the Charney-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. Throughout the thesis, our main tool is a new notion of cross ratio on Roller boundaries of CAT(0) cube complexes. We develop a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. We then study under what conditions such boundary maps extend to cubical isomorphisms. In the hyperbolic case, our results extend part of the work of Bourdon and Xie on Fuchsian buildings. Along the way, we also analyse the relationship between Roller boundaries, contracting boundaries and visual boundaries of CAT(0) cube complexes. For a Gromov hyperbolic group Γ, this allows us to embed the space of cubulations of Γ into the space of Γ-invariant cross ratios on the Gromov boundary of ∂∞Γ. This draws an unexpected similarity between cubulations and the many other geometric structures that can be encoded in terms of boundary cross ratios: for instance, points of Teichmüller space, Hitchin representations, geodesic currents.