Sublinear biLipschitz equivalence and sublinearly Morse boundaries

TL;DR

This paper introduces the concept of sublinear biLipschitz equivalences (SBEs) and demonstrates their invariance of sublinear Morse boundaries in proper geodesic metric spaces, with applications to group theory and random walks.

Contribution

It establishes the invariance of sublinear Morse boundaries under SBEs and develops the theory of sublinear rays, extending the understanding of geometric group invariants.

Findings

Sublinear Morse boundaries are invariant under suitable SBEs.

Sublinear rays generalize quasi-geodesic rays and are used in proofs.

Random walks on groups can be viewed as sublinear rays.

Abstract

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 $\kappa$, $\kappa$-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 $\kappa$-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 sublinear biLipschitz equivalence. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.