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Isometric actions on locally compact finite rank median spaces

Mohamed Lamine Messaci

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.

Isometric actions on locally compact finite rank median spaces

Abstract

We prove that a connected locally compact median space of finite rank which admits a transitive action is isometric to endowed with the -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.
Paper Structure (22 sections, 46 theorems, 42 equations, 5 figures)

This paper contains 22 sections, 46 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Median spaces and algebras
  5. Halfspaces and poc sets
  6. Duality
  7. An embedding of the convex hull
  8. Description of the convex hull
  9. Embedding lemma
  10. Characterization of compactness by mean of halfspaces
  11. Convex hull of compact subsets
  12. Proof of Theorem \ref{['local_compactness']}
  13. Transitive actions on median spaces of finite rank and local compactness
  14. Trace of halfspaces on convex sets
  15. Proof of Theorem \ref{['thm_principal']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be a connected locally compact median space of finite rank which admits a transitive action, then $X$ is isometric to $(\mathbb{R}^n,l^1)$.

Figures (5)

  • Figure 1: Any halfspace which separates $x$ and $y$ is either transverse to $C_1$, to $C_2$ or to the interval $[c_1,c_2]$.
  • Figure 2: The configuration arising in the second part of the proof of Lemma \ref{['depth_hyperplane']}
  • Figure 3: The segment $[x_0,x_1]$ is fixed by the action of the group of isometries of the real tree.
  • Figure 4: The orbit of $x$ in $C=T\times [0,+\infty[$ is non discrete but still the action of $G$ on $C$ is neither Roller non-elemenatary nor Roller minimal (nor minimal).
  • Figure 5: The subgroup generated by the inversions along the edges of the space $X$ acts Roller non elementarily and Roller minimally on the latter.

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: Helly's Theorem
  • Definition 2.3: Halfspaces
  • Theorem 2.4: Separation theorem
  • Definition 2.5: and notations
  • Proposition 2.6: See Proposition 2.3 Fior_median_property
  • Remark 2.7
  • ...and 82 more