  We characterize the Lie groups with finitely many connected components that
are $O(u)$-bilipschitz equivalent (almost quasiisometric in the sense that the
sublinear function $u$ replaces the additive bounds of quasiisometry) to the
real hyperbolic space, or to the complex hyperbolic plane. The
characterizations are expressed in terms of deformations of Lie algebras and in
terms of pinching of sectional curvature of left-invariant Riemannian metrics
in the real case. We also compare sublinear bilipschitz equivalence and coarse
equivalence, and prove that every coarse equivalence between the logarithmic
coarse structures of geodesic spaces is a $O(\log)$-bilipschitz equivalence.
The Lie groups characterized are exactly those whose logarithmic coarse
structure is equivalent to that of a real hyperbolic space or the complex
hyperbolic plane. Finally we point out that a conjecture made by Tyson about
the conformal dimensions of the boundaries of certain hyperbolic buildings
holds conditionally to the four exponentials conjecture.
