Large-scale sublinearly Lipschitz geometry of hyperbolic spaces

TL;DR

This paper investigates the properties of large-scale sublinearly Lipschitz maps in hyperbolic spaces, introducing a boundary invariant that distinguishes hyperbolic symmetric spaces up to sublinear biLipschitz equivalence.

Contribution

It introduces a boundary invariant for hyperbolic spaces that helps classify hyperbolic symmetric spaces under sublinearly Lipschitz maps, extending the understanding of their large-scale geometry.

Findings

Boundary invariant distinguishes hyperbolic symmetric spaces up to SBE.

Sublinearly biLipschitz equivalences induce boundary maps reminiscent of quasiM{"o}bius mappings.

Answers Dru ext{"u}'s question on classifying hyperbolic spaces via boundary invariants.

Abstract

Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly biLipschitz Equivalences (SBE) are a weak variant of quasiisometries, with the only requirement of still inducing biLipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of their boundary extensions, reminiscent of quasiM{\"o}bius mappings. We give a dimensional invariant on the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Dru\c{t}u.