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We show that a complete connected locally compact median space of finite rankwhich admits a transitive action is isometric to $\mathbb{R}^n$ endowed withthe $\ell^1$-metric. In parallel, we give a characterization of the compactnessin complete median spaces of finite rank by the combinatorics of theirhalfspaces. Under a mild assumptions on the isometry group, we show also thatall the orbits of a complete connected median space of finite rank arediscrete.

Isometric actions on locally compact finite rank median spaces