Actions des groupes topologiques sur les objets universels

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.