Infinite-dimensional Polish groups and Property (T)

Abstract

We show that all groups of a distinguished class of \guillemotleft large\guillemotright\ topological groups, that of Roelcke precompact Polish groups, have Kazhdan's Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group $\operatorname{Aut}(\mu)$ of measure-preserving transformations of the unit interval and the group $\operatorname{Aut}^*(\mu)$ of non-singular transformations of the unit interval. More precisely, we prove that the smallest cocompact normal subgroup $G^\circ$ of any given non-compact Roelcke precompact Polish group $G$ has a free subgroup $F\leq G^\circ$ of rank two with the following property: every unitary representation of $G^\circ$ without invariant unit vectors restricts to a multiple of the left-regular representation of $F$. The proof is model-theoretic and does not rely on results of classification of unitary representations. Its main ingredient is the construction, for any $\aleph_0$-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.