  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.
