  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.
