Actions des groupes topologiques sur les objets universels

TL;DR

This thesis investigates the existence of universal objects in topological groups and their actions, providing new criteria for amenability and showing limitations of certain universal groups in uncountable settings.

Contribution

It introduces new test spaces for amenability and extreme amenability, and demonstrates that the isometry group of a universal uncountable Urysohn space is not universal for its weight.

Findings

Polish groups are amenable iff all actions on the Hilbert cube have invariant measures.

Actions on the Cantor space can detect amenability and extreme amenability.

The isometry group of a universal uncountable Urysohn space is not universal for its weight.

Abstract

In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for amenability. We observe that a Polish group is amenable if and only if its every continuous action on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as amenability at infinity of discrete countable groups. As corollary, the latter property can also be tested by actions on the Hilbert cube. These results generalise a criterion due to Giordano and de la Harpe. In the second part of this thesis we are motivated by the problem of existence of an universal topological group of uncountable weight. We show that the group of isometries of a universal Urysohn-Katetov metric space of uncountable density is not a universal group of the corresponding weight. For instance, it does not contain the group of isometries of the classical separable Urysohn metric space. This stands in sharp contrast with Uspenskij's 1990 result about the group of isometries of the classical separable Urysohn metric space being a universal Polish group.