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.

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We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

We show that quasi-convex subgroups of negatively curved manifold groups with codimension one have nicely embedded limit sets in the visual boundary if the complement of the limit sets admits what we call strong barycenters, a property related to the absence of large diameter sets with 'positive curvature'. Furthermore, we show that the same result can be obtained if, in the complement of the limit set, simplicial complexes can be subdivided in a way that 'shrinks' them metrically. This provides us with two sufficient geometric conditions for the limit set of a quasi-convex subgroup to be nicely embedded.

We prove that the sublinearly Morse boundary of CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} cubulated groups with factor systems continuously injects in the Gromov boundary of a certain hyperbolic graph Γ. We also show that for all CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continuously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.