Infinite-dimensional Polish groups and Property (T)

We show that all groups of a distinguished class of large 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 Aut(mu) of measure-preserving transformations of the unit interval and the group Aut * (mu) of non-singular transformations of the unit interval. More precisely, we prove that the smallest cocompact normal subgroup G. of any given non-compact Roelcke precompact Polish group G has a free subgroup F <= G degrees of rank two with the following property: every unitary representation of G degrees 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 substructureswith suitable independence conditions.

Automated summary

The study shows that large topological groups within a distinguished class, Roelcke precompact Polish groups, have Kazhdan's Property (T) and provides specific examples of such groups.