Large-scale sublinearly Lipschitz geometry of hyperbolic spaces

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 Druc{t}u.