Infinite-dimensional Polish groups and Property (T)
Résumé
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 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∘ 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 ℵ0-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.
Origine : Fichiers produits par l'(les) auteur(s)
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