Infinite-dimensional Polish groups and Property (T) [article]

Tomás Ibarlucía
2020 arXiv   pre-print
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(μ) of measure-preserving transformations of the unit interval and the group Aut^*(μ) of non-singular transformations of the unit
more » ... l. 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^∘ of rank two with the following property: every unitary representation of G^∘ 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 ℵ_0-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.
arXiv:1903.00203v2 fatcat:mnxwrzrkjzcdjjnyqkfu5qrt3y