Quasi-isometric rigidity is often, though not always, obtained by proving that a given group has few self quasi-isometries. In general, the solvable Lie groups (and their lattices) have more quasi-isometries than their semisimple counterparts. Nevertheless, reformulating works by other authors we can show that for a class of "rank one" solvable Lie groups, including the three-dimensional SOL, the quasi-isometries are often (and conjecturally, almost always) rough isometries, that is, they preserve any left-invariant Riemannian metric on the group up to a bounded error. This property is sometimes enough to deduce quasi-isometric rigidity, after some additional work. This is based on joint work with E. Le Donne and X. Xie; I will also review parts of recent works of T. Ferragut and of T. Dymarz, D. Fisher and X. Xie. Link to the seminar webpage for more information - https://www.wwu.de/AGKramer/index.php?name=TSSS23&menu=teach&lang=de