Julien Melleray

Generic representations of abelian groups and extreme amenability

If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group

Polish topometric groups

\overline {\pi (\Gamma )}

GREY SUBSETS OF POLISH SPACES

for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic

Some geometric and dynamical properties of the Urysohn space

\overline {\pi (\Gamma )}

Linearly rigid metric spaces and the embedding problem

; in the other two, we show that the generic