*preprint*

**Inserted:** 21 dec 2018

**Year:** 2017

**Abstract:**

We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon \in \mathbb{R}^+$, each such space is $(1,\epsilon)$-quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1, C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1, C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation.