Hyperbolic projections and topological invariance of sublinearly Morse boundaries

TL;DR

This paper establishes the topological invariance of sublinearly Morse boundaries for certain groups, introduces a new topology on these boundaries, and provides explicit descriptions for hierarchically hyperbolic groups and graph manifolds.

Contribution

It introduces a new topology on sublinearly Morse boundaries, proves their topological invariance under group actions, and offers explicit descriptions for specific classes of groups.

Findings

Sublinearly Morse boundary is well-defined up to homeomorphism.

A new topology on sublinearly Morse boundaries is constructed.

Explicit descriptions of boundaries for hierarchically hyperbolic groups and graph manifolds.

Abstract

We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse boundaries of graph manifolds using their actions on Bass-Serre trees.