Tameness of actions on finite rank median algebras

TL;DR

The paper addresses tameness of actions on finite-rank median algebras by linking geometric rank to the independence structure of median-preserving maps into $[0,1]$, and derives a generalized Helly selection principle via Rosenthal dichotomy. It proves that for compact median algebras of finite rank, the independence number of the MP family equals the rank, and that continuous MP maps achieve the same bound in the finite-rank compact setting; this yields sequential compactness of MP functions and tame dynamics. It further shows that any continuous action by median automorphisms on such an algebra is Rosenthal representable (hence dynamically tame), and that Roller-Fioravanti compactifications of finite-rank median algebras are often tame, with applications to CAT(0) cube complexes. The work extends tameness from rank-one dendrons to arbitrary finite rank, and provides universal embedding results: every topological group embeds into the automorphism group of a compact locally convex median algebra via a free-construction approach in Appendix B. Overall, the results illuminate a deep connection between median convexity, combinatorial independence, and low-complexity dynamical behavior, offering tools for both structure theory and dynamical applications.

Abstract

We prove that for (compact) finite-rank median algebras the geometric rank equals the independence number of all (continuous) median-preserving functions to $[0,1]$. Combined with Rosenthal's dichotomy, this yields a generalized Helly selection principle: for finite-rank median algebras, the space of all median-preserving functions to $[0,1]$ is sequentially compact in the pointwise topology. Generalizing joint results with E. Glasner on dendrons (rank-1), we establish that every continuous action of a topological group $G$ by median automorphisms on a finite-rank compact median algebra is Rosenthal representable, hence dynamically tame. As an application, the Roller-Fioravanti compactification of finite-rank topological median $G$-algebras with compact intervals is often a dynamically tame $G$-system.

Theorems & Definitions (38)

• Definition 2.1
• Definition 2.4
• Definition 3.1
• Theorem 3.2: Characterization of Rank via Independence number
• proof
• Theorem 3.3
• proof
• Remark 3.4
• Example 3.5: Not finite rank but subinfinite
• proof