From homogeneous metric spaces to Lie groups

We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively. After a review of a number of classical results, we use the Gleason-Iwasawa-Montgomery-Yamabe-Zippin structure theory to show that for all positives, each such space is (1,s)-quasi-isometric to a connected metric Lie group (metrized with a left-invariant distance that is not necessarily Riemannian). Next, we develop the structure theory of Lie groups to show that every homogeneous metric manifold is homeomorphically roughly isometric to a quotient space of a connected amenable Lie group, and roughly isometric to a simply connected solvable metric Lie group. Third, we investigate solvable metric Lie groups in more detail, and expound on and extend work of Gordon and Wilson [31, 32] and Jablonski [44] on these, showing, for instance, that connected solvable Lie groups may be made isometric if and only if they have the same real-shadow. Finally, we show that homogeneous metric spaces that admit a metric dilation are all metric Lie groups with an automorphic dilation.