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

TL;DR

This paper discusses a recent theorem proving that all Roelcke precompact Polish groups possess Kazhdan's property (T), linking model theory, group theory, and metric structures.

Contribution

It establishes that every Roelcke precompact Polish group has property (T), extending previous work and providing a proof outline for the measure-preserving transformations case.

Findings

All Roelcke precompact Polish groups have property (T)

Characterization of these groups as automorphism groups of $\u2205_0$-categorical metric structures

Outline of proof for the measure-preserving transformations group

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.