The dynamical hierarchy for Roelcke precompact Polish groups

TL;DR

This paper explores the structure of function algebras on Roelcke precompact Polish groups using model theory, revealing that strongly uniformly continuous functions are weakly almost periodic and establishing links between tame functions and NIP formulas.

Contribution

It provides a model-theoretic analysis of function algebras on Roelcke precompact Polish groups, showing new relationships between various classes of functions and their dynamical properties.

Findings

Every strongly uniformly continuous function is weakly almost periodic.

Tame functions correspond to NIP formulas.

The isometry group of the Urysohn sphere is $ ext{Tame}igcap ext{UC}$-trivial.

Abstract

We study several distinguished function algebras on a Polish group $G$, under the assumption that $G$ is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of $\aleph_0$-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urysohn sphere is $\Tame\cap\UC$-trivial.