Sublinearly Morse boundaries from the viewpoint of combinatorics

Abstract

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.

Sublinear bilipschitz equivalence and sublinearly Morse boundaries

“…Furthermore, they are metrizable topological spaces [29]. Since their introduction, sublinearly Morse boundaries are studied and compared to Gromov boundaries in various ways, such as via visibility, divergence, and contracting properties (see [20,24] and [36]).…”