Title: From homogeneous metric spaces to Lie groups,
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.