Sublinear bilipschitz equivalence and sublinearly Morse boundaries

A sublinear bilipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function, the associated sublinearly Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that sublinearly-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi-geodesic rays. As an application, we distinguish a pair of right-angled Coxeter groups brought up by Behrstock up to SBE. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.