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We approach the quasi-isometric classification questions on Lie groups byconsidering low dimensional cases and isometries alongside quasi-isometries.First, we present some new results related to quasi-isometries between Heintzegroups. Then we will see how these results together with the existing toolsrelated to isometries can be applied to groups of dimension 4 and 5 inparticular. Thus we take steps towards determining all the equivalence classesof groups up to isometry and quasi-isometry. We completely solve theclassification up to isometry for simply connected solvable groups in dimension4, and for the subclass of groups of polynomial growth in dimension 5.

Metric equivalences of Heintze groups and applications to classifications in low dimension