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 pro-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 Roelcke precompact Polish group G (under a non-triviality assumption) 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.
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