We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as SLn(Z) for n ≥ 3. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call twostep representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.

中文翻译：

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高阶晶格的强属性 (T)

我们证明了高阶简单李群或高阶简单代数群在局部域上的乘积中的每个格都具有 Vincent Lafforgue 的强性质 (T)。在非阿基米德局部域上，我们还证明了它们对于所有具有非平凡类型的 Banach 空间具有强 Banach 性质 (T)，而一般而言，我们通过对 Banach 空间的附加假设获得了这样的结果。新颖之处在于我们处理非共紧格，例如 SLn(Z) for n ≥ 3。为此，我们引入了更强形式的强属性 (T)，它允许我们处理比群表示更一般的对象在我们称为两步表示的 Banach 空间上，即由我们只能组合一次的不同 Banach 空间之间的一组运算符索引的族。