  We consider Lie groups that are either Heintze groups or Sol-type groups,
which generalize the three-dimensional Lie group SOL. We prove that all
left-invariant Riemannian metrics on each such a Lie group are roughly similar
via the identity. This allows us to reformulate in a common framework former
results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent
results of Ferragut and Kleiner-M\"uller-Xie, on quasiisometries of these
solvable groups.

Group Theory;Differential Geometry;Geometric Topology;Metric Geometry