Property (T) for Roelcke precompact Polish groups (after Ibarluc\'ia, building on work of Ben Yaacov and Tsankov)

Abstract

We present a recent result of Ibarluc\'ia stating that every Roelcke precompact Polish group has Kazhdan's property (T). This striking theorem builds on the characterization of Roelcke precompact Polish groups as automorphism groups of $\aleph_0$-categorical metric structures due to Ben Yaacov and Tsankov. We introduce the underlying concepts coming from model theory, and we outline Ibarluc\'ia's proof in the (already new!) particular case of the group of measure-preserving transformations of a standard probability space.