Isometric actions on locally compact finite rank median spaces

TL;DR

This paper characterizes connected locally compact median spaces of finite rank with transitive isometric actions as Euclidean spaces with the ℓ¹-metric, and explores orbit discreteness under certain regularity conditions.

Contribution

It provides a classification of median spaces with transitive actions and introduces a new characterization of compactness in finite rank median spaces.

Findings

Spaces with transitive isometric actions are isometric to ℝ^n with ℓ¹-metric.

Under regularity conditions, all orbits are discrete.

Characterization of compactness via halfspace combinatorics.

Abstract

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. In the other side, 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 our way to prove these results, we give a characterization of the compactness in complete median spaces of finite rank by the combinatorics of their halfspaces.