Tameness of actions on finite rank median algebras

Abstract: In compact finite-rank median algebras, we prove that 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 Boolean-tame (e.g., finite-rank) median algebras, the space of median-preserving functions to $[0,1]$ is sequentially compact in the pointwise topology. Using the Bourgain-Fremlin-Talagrand theorem, we show that the set $\mathcal{BV}_r(X,[0,1])$ of $r$-bounded variation functions is a Rosenthal compact for every compact metrizable finite-rank median algebra $X$. 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 any finite-rank topological median $G$-algebra with compact intervals is a dynamically tame $G$-system.