  We discuss the notion of sublinearly bilipschitz equivalences (SBE), which
generalize quasi-isometries, allowing some additional terms that behave
sublinearly with respect to the distance from the origin. Such maps were
originally motivated by the fact they induce bilipschitz homeomorphisms between
asymptotic cones. We prove here that for hyperbolic groups, they also induce
H\"older homeomorphisms between the boundaries. This yields many basic examples
of hyperbolic groups that are pairwise non-SBE. Besides, we check that
subexponential growth is an SBE-invariant.
  The central part of the paper addresses nilpotent groups. While
classification up to sublinearly bilipschitz equivalence is known in this case
as a consequence of Pansu's theorems, its quantitative version is not. We
introduce a computable algebraic invariant $e=e_G<1$ for every such group $G$,
and check that $G$ is $O(r^e)$-bilipschitz equivalent to its associated Carnot
group. Here $r\mapsto r^e$ is a quantitive sublinear bound.
  Finally, we define the notion of large-scale contractable and large-scale
homothetic metric spaces. We check that these notions imply polynomial growth
under general hypotheses, and formulate conjectures about groups with these
properties.
