Isometric actions on locally compact finite rank median spaces

We prove that a connected locally compact median space of finite rank which admits a transitive action is isometric to \mathbb{R}^{n} endowed with the \ell^{1} -metric. On the other hand, replacing the transitivity assumption on the group of isometries by a certain regularity of the action on the compactification of the space, we show that all orbits are discrete. In the course of proving these results, we provide a characterization of compactness in complete median spaces of finite rank by the combinatorics of their halfspaces.