Sublinearly Morse boundaries from the viewpoint of combinatorics

TL;DR

This paper establishes a connection between sublinearly Morse boundaries of cubulated groups and hyperbolic graphs, providing a combinatorial description of convergence to Morse geodesics in CAT(0) cube complexes.

Contribution

It introduces a continuous injection of sublinearly Morse boundaries into hyperbolic graph boundaries and offers a combinatorial perspective on convergence in CAT(0) cube complexes.

Findings

Sublinearly Morse boundary injects into hyperbolic graph boundary

Convergence to Morse geodesics described via hyperplanes crossing

Subspace of Roller boundary surjects onto sublinearly Morse boundary

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.