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.

中文翻译：

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无限维波兰群和性质 (T)

我们证明了一类杰出的“大”拓扑群，即 Roelcke 预紧波兰群的所有群，都具有 Kazhdan 性质 (T)。这回答了 Tsankov 的问题，并概括了 Bekka（对于无限维酉群）以及 Evans 和 Tsankov（对于亲寡态群）的先前结果。进一步的例子包括单位间隔的度量保持变换的组Aut(μ)和单位间隔的非奇异变换的组Aut*(μ)。更准确地说，我们证明了任何给定的 Roelcke 预紧波兰群 G（在非平凡假设下）的最小协紧正规子群 G• 有一个自由子群 F ≤ G• 的秩为 2，具有以下性质：没有不变单位向量的 G• 的每个酉表示都限制为 F 的左正则表示的倍数。该证明是模型论的，不依赖于酉表示的分类结果。它的主要成分是，对于任何 ℵ_0 分类度量结构，自由群在具有适当独立条件的基本子结构系统上的动作的构造。