Sublinearly Morse boundaries from the viewpoint of combinatorics

Abstract

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.