Todor Tsankov

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S∞, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand–Raikov theorem holds for topological subgroups of S∞: for all such groups, continuous irreducible representations separate points in the group.

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

Decidability of Definability

\overline {\pi (\Gamma )}

Metrizable universal minimal flows of Polish groups have a comeagre orbit

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

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

\overline {\pi (\Gamma )}

Extremely amenable groups via continuous logic

; in the other two, we show that the generic